Lots of good responses here.
Most managers I know collect the funds from Section 8 and then pay out to owners on the same date each month.
One of the most annoying things about working with Section 8 is just the old school communication. They'll mail you to notify you of inspections. When they do that, they just give you a time frame of when they're going and the PM and tenant have to be there. Mentioning this because you definitely want to have someone who can go to those on short notice when you get your letter in the mail. If you go the self-management route you might be able to find a newer agent who is cutting their teeth in the industry and looking to make extra cash. Just pay them like, a flat fee or an hourly rate or something to help with in-person things.
I agree with @Drew Sygit about confirming operations when you're vetting. Each management company operates differently, so they're not all going to be a fit for you. I'd get a really good idea of what you want, and then just interview until you find that. But if you know you want to self manage and have the time to do that, then just develop a good relationship with a new-ish agent. Let them know you're looking for help until you can hire someone. You can do most of the work, and they can just be there for on-call work.
How were you burned by past PMs? Sounds like repairs and/or rent collection might be concerns of yours?