Yes, they will still move. Sending a letter to them has nothing to do with them moving.
One thing you can do that I'm pretty sure is safe is to time travel a bit until you get someone to ping about wanting to move. If it's someone you want to keep, tell them not to or vice versa if it's someone you want to get rid of. Regardless, once you do talk to them about their plans, you should be able to time travel to the desired time with either all of the villagers staying put or the one you want to leave being gone.
I feel like I did this method in the past to repair grass erosion that was making my town look pretty ugly, though I might be forgetting something as it has been a few months.
I can't speak to how safe it is time traveling backward though. The only experience I have with time traveling backward is going back to the last day I played after taking a break for a few days/weeks/months since I keep a log of my last date and time played. This is assuming you care enough to time travel back to the current time.
Alternatively, you could just buy the sets off of someone in one of the trade sections of the forum and avoid the time travel entirely.
Thanks! I'll have to just go day by day in that case (don't have enough bells to buy a set). I had a villager ping me about moving out the other day, but after I said "go!" they decided to stay...
I tt'ed back an entire two years and suffered no ramifications. Someone pinged to move, I stopped them, saved and then tt'ed from 2018 back to 2016. Only a day had passed. I think it's safer to go backward than forward.
I believe if you TT back in time, you can go as far as you want and it only counts as one day... it's travelling forwards that will cost you your villager... at least that's what I have heard and I have never lost villagers moving backwards (sometimes years)...
