The bad news is your team is terrible. The good news is you have multiple premium picks for multiple years, and that puts you in a position to draft BPA without worrying about filling holes. The other good news is that GM Nick Caserio shown skill at finding talent with his limited picks to date. With picks 67, 89 and 147, he found a possible franchise QB (which may be the steal in the draft, Micah Parsons notwithstanding), and a starting/rotational WR and TE. The goal should be to surround your QB with talent so that you can figure out if he is the real deal. That determination is hard to make when his offensive line and supporting cast are JUCO quality. If he is your franchise QB, you have a 3-year window where his salary is cheap enough that you can sign FA to put you “over the top”, so the click is ticking on this decision. If you commit to him too late, you may only have one year before his salary weighs the team down (like Dak Prescott), and you find yourself having to amputate crucial pieces to fit under the cap.

While rebuilding the offense, you should also pick the players that are falling due to medicals (Ojabo/Jameson Williams), or position surplus value (C Tyler Linderbaum), to maximize value. You do not expect to win this year anyways, so why should medicals bother you? Essentially, every position except for QB should be a possibility. Now, go have fun.
Pick 3
Trade. Down. This is a terrible year to trade down, and the return will not match the usual return. Still, the Texans need to fill 20 starting roles, and even the 7th and 36th picks would give you a star and a starter. Your best bet is if the Giants want a particular OT, and are willing to deal for him, or if they want CB Gardner and you can get them into a bidding war. However, guessing at trades is an impossible exercise, so we will assess the picks as they stand, with picks prioritize in order of mention.
DE Aiden Hutchison likely will not be there. DE Travon Walker still has to show that he can rush the passer. Drafting him this high is akin to drafting a QB that still has not shown accuracy, and would present too much uncertainty for such a premium pick. I love CB Sauce Gardner, but his traits are not needed for the Lovie Smith defense, and you don’t want to overpay for features you don’t need. Kayvon Thibodeaux is likely “not a culture fit” for Easterby. In keeping with the “help Davis Mills” theme, I would take an OT. Evan Neal is the steadiest pass protector, but Ikem Ekwonu is a run mauler. Laremy Tunsil is signed for three years, so you don’t necessarily need a LT, a RT will do. I would still go with Neal, as pass protection is the trait you are looking for with such a high pick.

Pick 13
WR Garrett Wilson should be the pick, but he is likely gone. If he is, the Texans should pick Jameson Williams. He may not be ready for game 1, but the Texans can wait and they will get enough games with him before next year to see if Mills can take advantage of his speed and separation. If he is gone, you are likely targeting BPA, even if not on offense. DE is a premium position and Jermaine Johnson should be available. The “culture fit” with Thibodeaux may not be as big a problem at 13, so I would take him at 13 as well. WR Chris Olave is likely available. This is a bit higher than his projection, but his excellent speed and route running create separation downfield. If Mills routinely misses open receivers downfield, this answers your question about your franchise QB. If he hits Olave, then you have your potential connection moving forward.

WRs Drake London and Treylon Burks should be available, and a “body receiver” would add contrast to the speed that Brandon Cooks already provides, thought 13 seems high for second tier WRs. The Texans have a gaping hole at CB, so Derek Stingley would roster talent at a premium position. The available IOL, LB and S do not present sufficient surplus value for this pick. Combine darling DT Jordan Davis is tempting, but he lacks sack production and doesn’t have short shuttle or 3-cone times to project pass rush abilities. You would be betting on size and 40-time, which is significant uncertainty for a high pick. Travis Jones is the only DT that can boast pass rush production and combine readings to show lateral and turn quickness, but this is too high for him. One of the foregoing should be available, given the number of teams needing edge defenders and QBs.
Pick 37
DE David Ojabo would be a top 20 talent at 37. IOL Tyler Linderbaum, Zion Johnson and Kenyon Green should be considered, in that order. Travis Jones would provide interior pass rush and run stuffing. LBs Nakobe Dean and Devin Lloyd may still be available, as might edge defender George Karlaftis if teams dislike his testing numbers. WRs Sky Moore and George Pickins would also present appropriate value and promote the “help Mills” train. Edge Boye Mafe would be a bit more of a project, but the Texans have time. I like DE Drake Jackson, but I do not think anyone else does at so high a pick.

Pick 68
If the Texans have not taken two edge defenders by this point, Drake Jackson’s testing profiles as the third best edge. 6’4 burner WR Christian Watson and Trey McBride would provide provide Mills with more receiving weapons, as would John Metchie (whose ACL injury they can wait on). There are a slew of IOL available here that could help keep the pocket from collapsing on Mills, including Ketucky’s Darian Kinnard, Memphis’ Dylan Parham, Chattanooga’s Cole Strange and Kentucky’s Luke Fortner. Solidifying that interior would help Mills with both a running game, and finally allowing him to throw without a defender directly in his face.

Pick 80
One of the picks mentioned above will likely be available. I am much higher on him than most, but TE Jelani Woods has the speed and agility of Travis Kelce with more length and a history as an in-line blocker. CB Tariq Woolen is a rangey, speedy corner out of UTSA that could help fill that gaping hole. CB Coby Bryant and WR Alec Pierce, while not instant starters, would be good value at positions of need. Edge defenders Dominique Robinson out of Miami(Ohio) and Sam Williams might also be available.
Picks 107 and 108
If Woods went before 80, Charlie Kolar presents the same size, but with slightly reduced athletic testing and less blocking. OG Ed Ingram out of LSU is available if the o-line redo hasn’t been completed by these picks. WR are littered throughout this draft, and post an easy “fall back position” if there are no other picks you value appropriate to the draft position. At this pick, you have Khalil Shakir, Calvin Austin and Danny Gray. OT Braxton Jones is a potential future starter as is the agile behemoth Matt Waletzko and Logan Bruss. For the time being, they could serve as a swing tackle.
Safety is also a consideration from here until the end. Safety is a position more reliant on processing speed than physical ability. .05 of speed is dwarfed by .3 seconds of faster recognition, and the avoidance of false steps. Since processing speed cannot be “timed” in the combine, many starting safeties with pedestrian test times and fast processing can be found.
Pick 183
It’s time to play the lottery. TE Chigoziem Okonkwo ran a 4.52 forty, and though he will never be a blocker, he can be a “power slot” tool. WR Velus Jones’ tape has routine explosion. Dane Belton and Juanyeh Thomas fit the mod of the quick-processing safety with unimpressive testing times.
Picks 205, 207, and 245
There are explosive RBs. Ty Chandler and D’Vonte Price are explosive RBs in the mold of Tony Pollard. (I like Chandler more). However, they need to be paired with a workhorse back like Keontay Ingram. Two of these picks could help settle the RB position cheaply, for the next few years. LB D’Marco Jackson is a spark-plug LB that would go two rounds higher if he wasn’t from Appalachian State. LB Brandon Smith is an intriguing prospect with the length/speed/quickness to profile as a potential edge defender.