As the others have said, Cozumle is not a place you want to make shore diving primary and boat dives as a back-up. The shore diving here is just so-so, unless you are into macro photography. Otherwise, you will get very bored, very fast and you won't see the diving that Cozumel is famous for.
The only true reef you can access from a shore dive is Paradise reef, and then only if the current is cooperating and you know how to navigate your way back. This can be a dangerous shore dive as well since it is right at the entry for the marina. If you have never been to Cozumel and plan to dive Paradise as a shore dive, I would highly recommend hiring a local guide, actually, it is required since it is within the marine park boundaries.
Otherwise, your choices are as mentioned above, but as someone above said, resort hopping is neither highly looked upon, nor is it practical...it's just not done here.
Villa Blanca (public, rent tanks from Papa Hogs)
Caribe Blu (public - basically the same area, slightly north - rent tanks from Blue Angel)
Fiesta Americana - If you are staying there - rent tanks from Dive House
Hotel Cozumel - if you are staying there - rent tanks from Dive Paradise
Scuba Club - if you are staying there - rent tanks from Scuba Club
Presidente - if you are staying there - ent tanks from Scuba-Du
La Ceiba - rent tanks from Aqua Worls or Dive with Martin IF you can get access through the hotel, which is doubtful these days
Schedule yourself for boat diving and use shore dives as a supplement...that's really the only way to do Cozumel.