This is always a challenge and why so many overseeding efforts are unsatisfactory.

Scalp the lawn before seeding. This is going to shock it and stop growth for a week or so. Go as low as you reasonably can without doing actual damage. I normally mow at 4", so I'd try to mow at 2". Which would probably take two mows and wouldn't be as much of a shock as I might hope, but my lawn is extremely thick and I wouldn't overseed, so...

Water like a new lawn. This is potentially going to cause fungal issues in your existing lawn. There is no way around this. Two to three times a day to keep the seed damp.

Hold off mowing as long as possible. In a fescue lawn, sprout should be in 5-7 days, with full sprout in 14.

When you do mow, be gentle. Walk slowly, move slowly, step carefully. Those sprouts aren't well-seated and try to step only on established grass. This mow will take you a while. If you slide on mud, you will destroy any sprouts you just slid on.

Mow at whatever height you like, but I'd probably resume at my normal mowing height. Then let it get very long again, for the first few mowing cycles. Fescue establishes fast, so you're lucky on that one. By month 2, the new fescue should be far enough along that you only need to be moderately careful.