How to tell if door latch is bad?

[tucsonjwt]

My driver's side door is hard to shut - have to slam it. I replaced the strike pin and bushing on the door jamb, which helped line the door up properly, but it is still had to close. I have about 1/4" -1/2" of play up when I grab the door and work it up and down. I expect that I need new bushings and pins in the hinges, but when I took it to a body shop I was told that I also need a door latch. The body man pushed the latch to the lock position and then showed me that it moved about 3/8" up and down when in the locked position. Does this mean the latch is bad? My passenger side door latch does the same thing, and it closes fine. He wanted $175 to replace the hinge pins and bushings and install a new latch, if I furnished the latch (hard to find a latch, he says.)