My lease ends on May 28, 2018 and I’m moving in with my uncle today (4/29) because it’s more cost efficient. I will still be paying the rent for the month of May as it’s the last month in my lease and I’d rather not have any issues with ending my lease early and I was just wondering if this is okay. A while back there was a leak in my bathroom ceiling and they came by to check it out. Since there’s going to be a new tenant for the month of June, I’m not sure if they’re going to come in and check it during May. I don’t want them to come in and see the place empty and think I abandoned it or something. Should I tell them? Or am I okay to move since my lease ends that month anyways?
I would tell them so that you can have your final walk-through and be done with it, surrender keys, etc. Then you don't have to think about it again. Your lease may have language specific to moving out early, but typically that's focused on notifying that you don't intend to renew. I've never seen language in a lease that would punish someone for paying rent but moving out a little early.
In some jurisdictions, the landlord is required to take reasonable steps to find a replacement when a tenant moves out early (they can't just collect rent on an empty unit for months on end without re-listing it) to mitigate cost to the former tenant. Given that it's just a month I wouldn't bank on recovering any rent, but if they find someone to move in prior to May 28 you'd be owed a refund (this may vary by jurisdiction as well).
So, there's likely no downside to letting them know you've moved already (check your lease), and the upside is that you don't have to think about the place anymore and there's an outside chance you'd get some of May's rent back.