I had the pet issue, too. Three cats and apartment searching. My ex agreed to keep "my cats" in the marital home until I found an apartment I could afford and allowed my pets.

In my area, there are a lot of apartment complexes that allow pets, even large dogs. Some even had dog pet playgrounds. I don't recommend living next to one of those. Too noisy but the dogs have a blast.

What you will likely find is they want a hefty "pet fee" upfront and it is non refundable. So, refundable security deposit, non refundable pet fee, first and last month's rent, were the main things expected. Some require documentation of insurance which is inexpensive and they will handle. Policy terminates when you vacate.

I politely asked them to waive silly stuff like $35 application fee and they did. Being polite to the leasing and maintenance staff during your occupancy will go a long way.

When you do find a place, make sure a staff person does a thorough walk-through of your new place with you present. There will be little things here and there....maybe a frayed area of carpet in a bedroom corner, a chipped wall corner, a scratch on the kitchen countertop. You also want to be present for the walk-through when you vacate and have your copy of the initial wall-through inspection.

Get them to notate that and keep a copy for yourself. They expect normal wear and tear but they can't ding you for replacing a countertop, for example, when the initial walk-through showed damage was already there.

I had to move four times during my divorce and three of those were apartments. PM me if you ever need support or want to vent.

PS: I applied as a single mother on disability and got approved no problem. Mainly depends on your credit rating. Talk to your attorney if you are having issues getting approved.