Its the original keyless entry. and that's what I thought too. So I chAnged the battery and that's when most of this started hAppening

 

Yes, it is a little game with mine. You have to run up to it and open or it will lock again. '97 with factory alarm. Also, I noticed the range on it seems short compared to the other vehicles I own. My Grand Cherokee will start from 2 blocks away. I have to be looking at my Z and talk nicely from over 100' for it to respond. I changed the battery in the remote with no change.