"no water heater, air conditioning, dish washer, maid service" - well, that would describe most places in Corozal. You would have a hard time finding any place with those luxuries. Also add no TV and no internet to your list. Even getting internet service and TV is often a problem (depending on the area). I don't think I've even seen a dish washer since I've been here.
There are several options, you can look at the real estate listings on line - they have homes for rent in with the houses for sale. Realtors like Remax and Century BTaL21 often post rentals. There are other property management companies on-line that have rentals. There is also craigslist and craigslist belize.
You will have to pay 1st and last month - plus one month security deposit on a long term rental. This seems to be standard. It may be tempting to by- pass that and find a place without a lease, but when you come to renew your monthly visitors permit, the officer can ask you for a copy of your lease( as well as a bank statement). It doesn't happen in other areas, but it is standard at the Corozal Immigration Office.
Utilities are usually extra - so factor that in when looking at rental prices. 2Mbps( which is probably 1/10 of your US speed) is about BZ$250 plus there are modem charges etc. If you have to have it installed, they charge non-residents a BZ$1000 deposit. Also, you may get sticker shock at your electricity bill.
There are several hotels that have long term apts for rent. The 'quoted price' is more usually expensive than a house- but they include utilities. This often makes them a better deal.
Hawk, who contributes here, often recommends the Maya Hotel, which is close to town.
Many of your options will depend on whether or not you plan to have a car or truck in Belize. There are allot of really nice, inexpensive 'off the grid places', but you need a good solid truck to get to them.
Just be careful about " work for rent". You need a work permit in Belize to do any "work" - even volunteer work needs a work permit. You have to be here for 6 months before you can even apply for one. It is done through the Labour Dept. The cost for a work permit is just over US $1000 and it must be renewed every year until you have your residency permit. If you come in on the QRP program, then you cannot work at all.
Booking into a short term rental for a few months will give you a chance to look around. You will also get better 'deals' when you know people here. From there you can find a longer term rental, then eventually something to buy. Don't be in a rush to jump into something - take your time.