Here's how we got around this problem and enable self-showings in occupied units:
1 - Set up an autoresponder when anyone inquires. This isn't possible for your average person to do with FB marketplace but for every other listing site it's easy.
2 - Tell the person that if they're interested they can book a showing. This could be either a self tour or not - if you're doing the showing yourself at least you will have avoided spending time trying to get tirekickers to setup a date. When you do this, also make the requirements incredibly clear.
3 - Get a smart lock setup outside for self-tours. Let the current, outgoing tenant decide when it's okay for self tours to happen. Most of the time they'll choose time windows when they know they won't be home. Some tenants would rather be there, in which case they'll usually choose less time windows in the week - not ideal for filling a unit quickly, but better than nothing.
If this is all setup correctly then you actually won't have to do much work between posting ads and reviewing the first applicant.