Many of them are second gear start by design. If the solenoid circuit is in good shape, you would also get first if you do a full throttle start.

To check the solenoid circuit, pull back the carpet to expose the kickdown switch. When the engine is running, there should be power going into the switch. If not, it’s the Klima or something further back: fuse or OVP relay. If there is power, then the switch itself, solenoid or wiring need a look.

Flare in third sounds like the K1 accumulator spring or piston, easy enough.

It would also be worthwhile to change the fluid and filter, you can do that when you swap springs.