How long does landlord have to send lease after deposit?

[Mattie Covatch]

Hello all, I would appreciate some info here on how long a landlord has to get a tenant a lease prepared after they apply.

I'm a tenant. One month ago, I applied for and was approved for an apartment at a large apartment complex. I immediately paid the security deposit.

A few weeks ago, they said the "only thing left" was for them to send the section 8 form in (request for tenancy). This complex regularly accepts section 8 and is known for section 8. It's a large cheap apartment complex in a rougher area of town. I have spotless rental history- lived in the same house for 10+ years. I recently got approved for section 8 after a lot of personal tragedies so I am looking for a place that accepts section 8 as my current place cannot/will not (too pricey). I have a steady job and clean background record with multiple forms of ID and pay stubs. My credit score is good. My downfall is the section 8 voucher.

I contacted them this week and now they're saying they're waiting for an apartment to open, then they'll send in the section 8 request for tenancy. Their website has shown that they have 20+ apartment units available right now for weeks. 

Is it likely that 20+ apartments showing as available now for weeks on their website really need renos/repairs before a move-in? Is it more likely that this apartment is stalling me?

I want to believe that they really have no units available right now even though their website shows 20+ units available now, and hope that I still have a chance at getting an apartment with them. But, I feel like they're stalling me. I don't want to contact them too often and annoy them, but am also concerned. They said they should have units opening later this week and would contact me after. I'm trying to patiently wait, but am worried that they're stalling me with no intentions of ever letting me move in.

Does an apartment complex landlord have to refund my security deposit paid already if they don't send me a lease after a certain amount of time?  I'm trying to gauge whether I should continue to look for other possible places or not. Thanks for any information anyone is able to share, I appreciate it.