I'm posting this for anyone who may see this issue in the future. I no longer have this car (thankfully this was the worst problem I experienced with this car) but back in 2016 I was leaving my job one summer and after starting it I went to shift into drive when I noticed a loose feeling on the gearshift. The car did not move and acted as if it didn't do anything. Initially I believed this was the dreaded transmission failure that plagues the final generation of the 626 model. After some sleuthing I found that a grommet that held the linkage cable onto the gearshift had broken off and disconnected it.

It took me an hour to get it back on thanks to the wonders of free information on the internet but a more permanent fix took another hour. My "fix" is more of a modification that was devised by someone on the Mazda626 forums of installing a cotter pin in place of the grommet which worked up until I sold the car. For those less inclined to work on their own vehicles you'll need to tow the car to a repair shop or dealership where they will either replace the cable or the grommet.