You say you can "hear water dripping into the drip catcher" ?
That would indicate that you unit is defrosting.

When you replaced the thermostat, did you completely de ice the evaporator?
If you had ice built up and did not clear all the ice your air flow could still be restricted because the heater cannot completely remove the excess ice, and air flow is reduced and not cooling properly.

Your most likely situation is the doors are not closing and sealing properly and room temperature air is entering the compartments. You need to check and make sure the doors seal all the way around the openings,especially the bottom of the doors.

You may also have a defective temperature control(thermostat) but at this point I doubt it, and my other recommendations should be done first.

Phil,

No, door seals don't normally go bad all of a sudden, unless the doors are knocked out of alignment, or the Door is overloaded. But it takes a while before the user notices it, by then they need to be replaced.

Did you remove the lower back panel on the fridge and clean the coil assembly? if not, you need to.

You should also remove the control panel and tunnel in the fresh food compartment ceiling, and check the damper assembly(styrofoam box)for ice or debris in the tunnels restricting the air flow to the temperature bulb of your control thermostat.

If you have done all of the above reccomended checks and repairs, I would have to think your temperature control may be malfunctioning.