There are some condos for rent not far from the Money Bar. You might look into one of those, although it might be pretty pricey to rent one on your own. There's a nice small hacienda style hotel called Village Tankah about 1/4 mile north of the cruise ship pier. I've stayed there dozens of times over the years and I like it quite a bit. There is no beach there.
Beaches are generally south on the island, where the big all inclusive hotels are. If that's your style, then try one of those. There are no 'house reefs' in Cozumel, all the standard dive sites are in the marine park and you must get on a boat with a licensed guide to visit them. There are some ok shore dives between the cruise ship pier and downtown; one of the more popular ones stretches from Villa Blanca hotel to Blue Angel. Either of those places might be a good option for you too.
If you want a really great beach (IMO) then Ventanas Del Mar, over on the ocean side of the island, is something special. The down side is that it's very isolated from the rest of the island and it's a long drive across the island to get to any of the dive sites and operators. A rental car would be an absolute must unless you like spending $100/day or more on taxis.
You didn't say where you are staying in Tulum, but if its in the beach zone, I don't think you'll find anything on Cozumel comparable. You might do best switching gears in that case and focus on diving.