I started this from Belize before re-reading your post. So reverse these first two sections:
You should see reef sharks in Belize pretty regularly. Boat diving in South Belize is all about an hour out on the barrier reef so prices reflect that - farther trips can be as much as $250 or so. Although you can dive closer in, the price reflects the distance also. No shore diving in that area - it's too flat and sandy. There is some cheap lodging in Placencia - one small waterfront hotel that doesn't have A/C seemed pretty reasonable. Moving farther north gets more expensive. Splash Divers lists a few options on their website.
Another option is Ambergris Cay. There are some close dives there and they also do some of the better atoll dives. Ambergris is about as far north as you can get and still be in Belize. Maybe check out Caye Caulker also - it's between Belize City and Ambergris. There's a ferry that runs between all 3.
You can take a bus to the border and cross over also. There's also a ferry to/from Puerto Cortes that is an option. It docks somewhere in south Belize. Belize is supposed to have good bus service also.
From there you can get to La Ceiba via bus. Getting to Utila since you're backpacking is very reasonable. Get to La Ceiba and take the Princess Ferry - it's $21.
Utila Princess - Utila, The Bay Islands, Honduras
Utila has a lot of low-cost accommodations since they do a lot of DM training. Dives are the same as Roatan - around $30 or less. Might be good to bundle that with your AOW and see if someone cuts you a deal.
There's a hostel there called Sea Side - slightly out of downtown. IDK if they book diving pkgs. with anyone but Alton's is often recommended. We stayed at an AI resort so IDK much about town since we only went thru it.
Sea Side Utila - Hostel Accommodation - Utila, The Bay Islands, Honduras
From Utila you can fly over to Roatan but it's pretty expensive. Day flights only there's no lights at the Utila airport. Worldwide Travel has a Facebook page - Alice lists available flights/prices there.
Or take the Princess back to La Ceiba and next door is the Galaxy Ferry terminal - they synchronize the schedules so you can get between the islands in under 1/2 day. We do it to get from Roatan to Utila and back since we fly into Roatan. The whole trip, both ferries costs $100 r/t.
If you need to fly to Honduras - it's generally cheaper to fly into San Pedro Sula and then make your way to La Ceiba. Or even fly direct to Roatan then on to Utila - it's not much more when you add up the ferry prices, bus ticket etc. Don't overnight in San Pedro - it's really very unsafe.
You can also fly from Belize City via TropicAir directly to Roatan if that works for your plans. But it's $175ish.
The Airline of Belize - Tropic Air
Besides what Joe mentioned above, there's another hostel in the West End.
nosotros
I'd stay at Chillies though if they have space - it's just across the street from the beach. Native Sons is their on-site dive operator. Buena Onda is a block or so inland. Some good general West End info is here:
Eating In/Out | Scuba Diving | Roatan, Bay Islands, Honduras | PADI
There's several small markets and daily local food truck vendors. They sell nice stuff, we bought produce from them a couple times. Cheap eats are the Noodles Factory or sandwiches at Sundowner's beach bar. Rotisserie Chicken (has a different actual name) over by Georphi's is very reasonable also. Tong's Thai is a little nicer but fairly reasonable also.
Consider renting a small cabin/apt. there. Some like the studios at Posada Arco Iris go for $50 or less and have small kitchenettes. Some of the 1BR cabins at Coconut Tree have full kitchens also - my friend rented one. I think it was $45/nt.
Almost all the nightlife in that area is in the West End. There's a water taxi between it and West Bay also.
Even a couple of shore dives but finding someone to rent you a tank can be difficult. If you do, they're #'s 8 and 9 on this list - the rest of them are at the AI resorts.
Scuba Shore Diving Site Listing for: Honduras, Roatan, Caribbean I've heard Sueno Del Mar might have rentals available but no one has ever confirmed that.
I don't go to Cozumel so don't know much about it. Some friends stayed at Villablanca Garden Hotel - they bundle diving with one of several dive operators on the property or next door. I think my buddy paid $800 for the week plus lunch/dinner - they have single occupancy pricing. It's a long walk to town though - I think he said 2? miles.
We also have sub-forums for all these locations so you might post specific inquiries there as well.
Bay Islands
Belize
Cozumel