You would absolutely have to at least get started with pool and classroom at home before your trip. Most certifying agencies have a maximum number of training dives you are allowed to do in a day and the number is less than the total number of confined water dives and the total number of open water dives you need to get to OW Diver. That is, you need at least two days of pool/classroom and at least two days of open water checkout. 4 > 3 days of vacation you are planning.
In addition, you have a 3 day, 3 night trip, which really means 2 days, 3 nights because you won't likely arrive early enough in the morning to do much. Even more, you need to allow yourself 24 hours after your last dive before you leave on a plane , so if your trip is Monday - Thursday (MWF nights), arrive on Monday at noon, you can dive on Tuesday (maybe Monday if you find somebody willing to work with you late) and Wednesday, but only if your flight home is in the afternoon on Thursday and you complete your dives early. Even trying to cram in checkout dives in that short of a trip is going to be difficult and you won't get the chance to do any fun diving.
I think you are also limited to only 3 adventure dives per day for AOW, so you would need another 2 days for that.
My recommendation would normally be to try to do your certification in whatever mudhole of quarry your local shop takes you to, and then enjoy how much better Cozumel is than that. But, since it's already late October, you might not be able to do a checkout near Pitt in the next month because of water conditions. My shop (in Iowa) has already finished our OW checkouts for the year and we won't start doing them again until May, but we do pool/classroom year round and send referrals for checkouts. In your case, I would actually recommend just enjoying what you can of Cozumel, which is still a lot, and if you get the urge to do it again (you will) get certified and go back as a diver next year. I fell in love with Cozumel when I went snorkeling there and I got certified and went back as a diver the next year.
ETA - Many shops will do what's called a "Resort Course" where they do some minimal in water training with you and then let you do an actual, closely supervised, shallow dive on scuba in open water. You don't have to be certified for that and you won't get certified as a result, but it's a good 1-day option for you. It's also a low cost way to find out if scuba is something you really want to do (maybe $150 vs. up to $1,000 for a full cert).
PADI has a class called discover scuba diving, it allows you to see if scuba diving is for you - if you’re unsure or have any doubts, definitely more cost effective route cause if you don’t like it, you’re out only that minimal cost. Discover scuba training can carry over to PADI open water training, doesn’t really save you much time at all as the bare minimum is covered in discover.