A slow drip in water dispenser

This sounds like a sticking solenoid valve. When you depress the lever for you water dispenser it activates a valve, usually at the bottom back of the fridge. It might look something like this:

Sometimes they just burn out and stop working or the seal inside fails sometimes from particulates in the water. Some people put a pre-filter on the water hose that attaches to the fridge just in case. If you have a water filter inside your fridge this does not protect the solenoid. They are next to impossible to repair so a replacement is needed. Just go to Amazon or eReplacement parts and search for your fridge model number with the search terms "water dispenser solenoid". If you are mechanically inclined you can replace yourself or get an appliance tech to do it. It's a 15 minute job.