Browns fan here, but wanted to chime in since we are all familiar with each others squad and have some of the same roster needs.
QB: if you can get a 2nd, 3rd for Andy i would do that in a heartbeat! anything beyond that and i would just keep him. Have him start the season as QB1 and let Joey get acclimated to the pros a bit. Also, you guys will have a "new" LT in Jonah Williams and presumably 1-2 other new OL acquisitions, so while it takes time for them to gel you probably don't want JB in there taking shots right off the bat.
RB: i've always liked the Mixon/Gio combo a lot, and with a stronger OL i think they can really shine as long as they stay healthy
WR: Tagging AJ is smart, but i hope for your sake if the trade deadline is close and you are not looking at a playoff spot you can at least cash in and trade him for a mid-round pick. Someone mentioned DPJ in round 3, but i think he will be gone by then. Boyd is good, Ross can help stretch the field, but after that the pickings are slim. Trust me, as a Browns fan i've known what it's like to have 1-2 good WR's and then nothing after that. this draft seems pretty deep at the mid-level WR spot so i think you'll find something there. Or in free agency
TE: Assuming you guys take Burrow, i wonder if you'll follow up with his TE from LSU (randy Moss' son, forgot his first name) since they have a good rapport. I think he is projected in the 4th-6th rounds so why not?
OL: Both of us will be in the market for most of the same OL. I have been reading that Billy Price is getting closer to a bust label, which is not something i ever thought i would see. Hopefully he can turn it around for you guys!
Defense: just lumped this all together to save time, but i think you need either defensive end or maybe a stud LB at the top of round 2. Of course that pick might be very valuable if certain players fall out of the first round. you aren't trading the #1 pick unless you get an insane package, so this might be where you pick up some extra picks. If i recall correctly CB and Safety aren't bad spots for you guys and an upgraded pass rush will certainly help both areas!
I see you guys spending a fair amount in free agency and i would not be the least bit surprised if both of us are battling for 5-6 of the same guys. we'll both have to overpay a bit to get some people in, but with the right moves (and some luck) i think both of us can improve by 3-4 wins next year.
It's certainly going to be an interesting free agency/draft period!
Good luck guys!