You could just enter new servers mysql login info to old server and work from both servers on new DB. Old server would be slower a bit due to mysql request have to travel long way, but it would be a few times easier and faster.
Yeah that, and write the redis data to the new server because it's not only for caching. But yeah it's a very viable option ! I'll add it.
One cool trick is that, after you've copied all the data to a new database, you offset the auto-increment to some larger value on all tables in the new DB. Then you can switch the old app to use the new DB and it will immediately start inserting new records into it, but will have a gap between the old and the new auto-incrementing values. If your app was getting a lot of writes you've probably ended up with a few extra rows in the old DB since it took you some time to copy the data over before you could switch the DB connection. However it's no problem since you've created that gap, so you can just copy the missing rows into the new DB later, there will be no collisions.
That assumes that all the inserts and auto increments happen in the same order for both databases, which can't be guaranteed. You might end up with users with different IDs, which is pretty bad.