2. Is this even a good idea? I worry about finding somewhere to stay without a reservation, and also slightly concerned we wont run into anything fun going without a destination.
Also if anyone wants to recommend road trip destinations anywhere in the continental US that would be incredibly helpful.
I've done a few road trips before, although none were really "spontaneous" in the way you are describing. But, I have been on road trips where I woke up in the morning, looked at a map to determine roughly how far I wanted to go that day, and reserved a hotel room for that night with no problems.
For your question: this is a good concern, but not as big of a fear as you may think (depending on how risk-averse you are I guess). You can also do something more spontaneous by 1) doing most of the trip mid-week so you're not competing for hotel rooms with weekend vacationers and 2) considering camping at campgrounds, which are EVERYWHERE, and cheap, and can be fun.
For trips: I almost spent a summer in Nashville and really wanted to do a day/overnight trip down to Muscle Shoals. You could probably hit the gulf pretty easily too. You could also drive into AR and do some driving and exploring in the Ozarks - beautiful place there with lots of camping options. If you enjoy cities, I did the eight hour drive from Nashville to Chicago once (actually a little north of Chicago) - there isn't a whole lot to see along the way aside from Louisville and Indy, but it's a pretty easy drive and obviously Chicago has tons to do.
I've done a number of cross country drives (VA to ID, ID to MD, VA to TN to MN, VA to CO, CO to MN, CO to TN and back), including several with my ex that were sort of romantic getaways, so feel free to message me for ideas or with other questions.