Craftsman lawn mower?

I have a craftsman 15.5 HP riding mower with a kohler CV15s
engine. The problem started with what I thought was a bad starter. When you turned the ignition at first the engine would
hardly turn then slowly the flywheel would gain speed until the
mower finally would crank. I replaced the starter but the problem was still there at first .Now when you turn the key nothing happens. but I can crank the mower while sitting in the seat(here is an artomatic cutoff switch) by placing a screwdriver across the terminals at the solenoid. What do I check next? Thanks for your help.