Model year would be helpful; specifications differ somewhat between them.
Picking a time stratum as a starting point, the US and California main jet size stock is 118 (previously, 120 for US); pilot screw out 2 turns (kinda ambiguous in Kawasaki KLR250 supplement service manual, could be 1-3/4, depending upon how countries are listed).
I concur, sounds like the carb is jetted 'way too rich.
While I have the 22-cent mod on my KLR650, IMHO, the KLR250 don't need no stinkin' 22-cent mod; YMMV.
The "choke" (starting enricher) plays its role with my '03 KLR250. After sitting a long time with no engine activity, I find turning off the fuel petcock and applying the choke helps on initial start-ups.
And, I confess; sometimes, starting fluid is my friend!
Be sure the diaphragm fabric is intact, and its peripheral edge fully seated in its groove in the carb body on reassembly. Nothing against Sea Foam, but . . . I'd favor a harsher chemical for a carb soak.
YMMV, but I find my KLR250 screams and runs like the wind with stock carburetor, intake and exhaust.