The button would need to be pushed to go from Neutral to Reverse to Park and again to get it out of Park into gear. Button action isn't needed between Neutral and 1.
As for slipping, how is the ATF level and condition? Has it had any recent ATF and filter change?
If the shifter moves by pushing on the gas pedal, then the kickdown shaft is binding with the concentric (one inside the other) gear range selector shaft on the top of the transaxle.
This was common in damp climates and with the battery mounted above this area, battery acid could mist down and start the corrosion. Once this kickdown lever binds, shift points can be late and variable. The kickdown bell-crank on top of the transaxle should move freely with a light spring return to the idle position. The kickdown return spring should be at the other end of the kickdown cable at the throttle body. Sometimes this spring is missing.