This year with the corona things might be a bit different but never had an issue just showing up at parks (maybe call in the morning and try to arrive early afternoon)...unless you are trying to make the trip over a holiday or there is a big event going on I wouldn't expect a problem finding a campsite.
Overnighting in a big box store parking lot...we've done it a couple times in a pinch but really not a fun way to travel. That said, common courtesy is to go into the customer service and ask if you can park overnight...if they say yes and you aren't causing problems, I wouldn't expect the police to come by and kick you out.
We just look out our preferred travel distance and google campground to see what's around that area. Also, if you are just overnighting and moving on in the morning, they likely will have more flexibility as they don't need to find a single site that's open for multiple nights. Some places also have overflow lots that may have fewer amenities but at least you are secure.
