I keep a log of my movers for just such reasons. I won't time travel until I've found the villager who wants to move and I either accept or deny the request. Once you've verified that, it's safe to travel. You only need to go to one date in the future and then back (rather than day by day). Check for movers on the day you travel to and then check for movers when you set the clock back.
It's not for everyone, especially if you're worried about favorites leaving, but with practice and sound advice from those who know what they're doing, it works: no one leaves (unless you want them to) and the grass comes back. Play for a bit, see if you feel the same about the grass or whatever, then decide if you want more info on how to time travel. I was scared at first, but once I got the hang of doing it safely, the game became so much more enjoyable.