Ok, so I got the pesky pin out. It was really just a leverage issue. I ended up lifting the vehicle up higher (safely with jack stands of course) so that I could get more umph on it and I got it on the first pull with a larger pair of pliers. The good news is that my problem was resolved after replacing the neutral safety switch. The vehicle now shifts properly in all gears. The error code was P0705.
The job was rather difficult and part of it was my fault. You have to drain the transmission and pull the pan to get to that pin I was talking about. Once the pin is removed, you can gently pry the shift linkage away from the transmission enough to release the neutral safety switch and remove it. You just need enough space to disengage the pin from the shifter that changes the gears on the transmission from the little slot on the neutral safety switch. Its connected right there and you can remove the neutral safety switch once you disconnect the linkage. Of course, you also need to remove the 3 8mm bolts holding the neutral safety switch onto the transmission.
I spent too much time farting around with that pin instead of using common sense. Then I spent a good chunk of time trying to connect the two connectors from the neutral safety switch into the wrong plugs. I thought they needed to go up in the back of the engine bay on the passenger side and I messed with it for awhile and did a bunch of unnecessary research. I watched a YouTube video that showed the connectors connecting in the back of the engine bay on the passenger side and the FSM showed that there were AT connectors there. It didn't make much sense for them to connect there because the cable seemed too long for that and one of the plugs wouldn't fit. Anyway, I dug deeper into the FSM and found that there is a listing for neutral safety switch on the harness diagram. The correct section of the FSM is EL-325 and the plugs are B65 GY8 and B66 GY2. The B## is just the location on the diagram and the GY8 means the connector is grey and has 8 cables coming into it. Thus GY2 would be grey with two cables going into it. The cables actually connect under the rear passenger side of the vehicle above the exhaust. There is a little panel you can remove to get to the connectors up in there. Thanks so much for the FSM Nico Club, it was super helpful!
One more helpful tip on the install. The neutral safety switch is adjustable and if you don't adjust it right you might have trouble with it shifting properly. There is a hole in the lever on the neutral safety switch that lines up with a hole on the metal housing of the neutral safety switch. Mine had two holes in the lever, one at the top and one toward the middle of the lever. I used a 3mm allen wrench to hold the lever in place so that when I installed it, it lined up properly. First get the pin from the shifter engaged with the lever of the neutral safety switch. Then install at least two of the three bolts into the transmission housing, but don't tighten it to much so that you can move the neutral safety switch around a little bit. Adjust it until you can put the short end of the allen wrench through the hole on the neutral safety switch lever through to the alignment hole behind it on the body of the neutral safety switch. If you use the short end it kinda sticks in there and doesn't get in the way. Once you have it there, tighten at least two of the three bolts that hold it to the transmission body tightly. Then tighten the third and remove the allen wrench.
You might ask why I didn't trace the old cable in the first place. Well, it was really hard to tell where it went up in there. It's all greasy and dirty and hard to see and follow. I actually ended up just cutting the cable off of the old neutral safety switch to remove it. I didn't plan on that, but it was damn near impossible to remove the old cable and not worth the effort. I just left the old cable up in there when I replaced it.
I didn't have to remove the cat or any exhaust parts to get to it. I ended up using an impact wrench to remove the plug in the pan. I know you aren't supposed to do that, but the plug is really flat and it's hard to find purchase with a typical socket or wrench. It was on there super tight and I was going to round it off with the ratchet. Plus, I didn't want to drain it by dropping the pan. It would have been a huge mess.
It ended up taking 6 quarts of ATF to refill afterward. I wasn't expecting it to take that much as it didn't seem like that much came out, but I guess it did.
This post isn't meant to be a how to, but I hope some of the info is helpful to someone down the line. Here is a link to the YouTube video that I watched. It isn't great because it doesn't show the full removal and replacement, but I thought it was helpful. Just be certain to check where the cables from the neutral safety switch connect to your specific model.
https://www.youtube.com/watch?v=OQ4u6AAt1As
Thanks to everyone who commented on this thread! I appreciated the help.