1. Would it be possible to still obtain an adoption spot reasonably in the next couple of hours if I started time traveling around?
Yes, that's possible. There are some methods that are pretty safe, too, if you're worried about losing villagers you like. It sounds intimidating and it does require some patience, but it's not as complicated as it sounds.
2. I would personally like the date to be set back to the original and is that possible without majorly screwing up my town? I have villagers I would like to keep but I don't understand how exactly backwards TTing affects moving patterns.
That's definitely possible because when you time travel, the bigger issue is when you're going forward in time. When you move backwards to the original date, that counts as only one day forward in the game so it's not the issue. If you're moving out a villager and want to adopt them out, when they tell you their move out date in the future, just do the maths and figure out how many days that is. When you go back, subtract a day and calculate the new date. For example, if a villager said they'd be moving in 5 days, when you go back they'll move out in 4 days instead.
The worst effects you can get, if you're careful, are some weeds and bed hair but those are easy to fix. Just make sure you have the Beautiful Town ordinance so your flowers won't wilt.
I'll tell you the method I use, if it helps. It's really safe, and I use it when I want to move someone out without losing my other villagers.
Ideally, you'd get someone to ping you, tell them to stay, and then set the 3DS calendar to a Monday of a week without events coming up. In my experience, though, if you're really desperate you can use the week you're on (as long as there are no events soon), or check ahead to the Monday you'd travel to using a new character (similar to what I'd explain below, but the first time you'd time travel and check ahead would be between your current date and that Monday instead of two different years, so ideally it would be over a week from now).
Load the game as your mayor and immediately save and quit. Set the 3DS calendar to one year from now. Go back to the game but don't load as your mayor. Instead, start as a new character. After you get the map from Porter, don't go to the town hall or anything. Just look at the map and check if anyone is missing. That missing villager will be the next to move out (if nobody is missing, you can safely load as your mayor on that date, as far as I know. It means that nobody is going to move at all).
Quit without saving and set the calendar back by one year again (so that it's on the date you last played as your mayor). Load the game and go find the villager who would be missing in the future. If they don't ping you, talk to them 5 or 6 times, save, quit, reload, and go find them again. Keep doing that last step until they ping you or tell you about moving.
If it's someone you want to keep, tell them to stay and save and quit. Go to the calendar and set it one year ahead again. This time, load the game as your mayor. Nobody will have moved out because you denied a request to move before time travelling. Save and quit. Set the calendar one year ahead, check who's missing using a new character, quit without saving, go back one year to the date you last saved as your mayor, and use your mayor to talk to the villager who would be moving as explained above. Keep doing this until it's a villager you want to move out.
If it's someone you want to move out, tell them to go. Pay attention to the date they give you and calculate how many days that would be from the date you're on. Save and quit.
You can now set the calendar back to the original date and load as your mayor. It'll count as one day forward, so the villager will be moving in the amount of days they told you before, minus one. Figure out what the new move out date is.
That's it. It sounds pretty complicated but it's easier once you get the hang of it.
