I am getting a divorce and still have a signed lease agreement. Should I still receive the BAH until the lease is done with?

Both are inaccurate answers. Being a SGT does not automatically entitle BAH. Also, the BAH does not "belong to the spouse." BAH is an allowance given to support the living arrangements of an entitled individual, who's status is deserving based on regulation. Your local JAG office will concur. There is a base level of spousal support that IS mandated, and the JAG can assist you with find that information.
I digress, the initial question. You are receiving BAH as an entitlement based on your marital status. Your entitlement ceases at the moment the divorce is approved, unless an exception is made as approved by your appropriate level commander. Being a government entity, it unfortunately does not adjust entitlements based on choice. The overly harsh answer that I have seen too many times is that both the divorce and the lease were choices, and you should have made plans accordingly. Not my circus, not my monkeys. Sorry to hear about your situation, but also would hate for you to have bad info. Go see JAG and finance, and include your chain in the discussion.

As soon as you get your divorce finalized your BAH with dependents will stop and depending on the base policy with single service members drawing SINGLE BAH. You may have to request it if you are still living out in town.

You're a SGT so I don't see why you would stop receiving BAH.