It sounds like the run capacitor is going bad. They are cheap enough that if you do not own a $50 amp meter to test it, you can buy a new capacitor for about $15, and just replace it.
Also check your voltage. I bought a Kill-A-Watt at Amazon.com and it will monitor the voltage. If it goes below 110 while running the A/C unit, it would explain the slow starting. It is not un-common for the voltage to drop a bit will a large load is on the line, but to much is not good. Many RV parks with only 30 amp receptacles - the receptacle gets overheated, and can be the cause of voltage drop, see if the plug is cool when not on, and if it gets warm when running the A/C. Mine was hot like a toaster at one RV campsite. I changed to the 50 amp receptacle with my 50-30 amp reducer, and everything returned to normal voltage, and I put away my voltage booster.
You might also try a super booter kit. IT is a start capacitor with a resister built in that heats up when to many amps are going through it, so it takes the start capacitor out of the circuit. They make them for 110 and 220 volts, get the 110 volt model.
Good luck,
Fred.