Deposit your check in a bank or credit union. CapOne, Chase, your local credit union, Ally, whatever - the exact institution doesn't really matter, but once it's deposited you won't have to worry about losing the check or letting it expire. (Not that it would be a disaster, you'd have to get the old check cancelled and a new one issued, but that's a hassle.)
You keep saying the hotlines can't help you, but I haven't seen any evidence for that assertion.
Great job talking to your coworker about apartments. This is a thing that normal people do and it is helpful because people can share knowledge (and their feelings, too). Keep it up!
I second the recommendation to ask church leaders or volunteers for advice as well, if you're going to a church. You are not the first person in the world to be seeking an apartment! People can have helpful advice.
I would ask if you're interested in following any of your old coworkers into new jobs, but it sounds like your priority is getting an apartment before a new job, which is a perfectly reasonable priority to have! Something to think about once you have a place.