My 1966 F-250 4x4 was used only to plow snow on a private road for years and years. The transfer case was never shifted out of 4L. The shift lever feels rock solid with no movement all. Do I need to push down or any other movements to get the lever to move? This is a Dana 24 transfer case. I can't move the shifter at all.

 

Nope. You should just have to pull it towards the seat. I would unhook the linkage, take the shifter **** off, unbolt the lever from the transmission (two bolts), and pull the lever out so you can try to get it freed up. Apparently nobody ever greased the zerk on the lever pivot point.

 

Parts catalog pic: 87913-S is the Zerk (grease) fitting at the lever pivot point.

 

I crawled under he truck for a look-see today and I think the problem is at the transfer case. There are two control rods that come out of the case connected by a bar. A shaft runs from that bar up to the lever. The lower rod has a little in and out movement to it but the upper rod seems solid. I tapped gently on them, and tried to gingerly pry the top rod out. But it seems stuck. Would starting the engine to spin the gears help. Maybe that would line things up a little better.
I've got the exhaust off right now so I'll have to wait until that's back on.
The case I'm guessing should have gear oil in it. I wonder if it leaked out and the shaft the gears slide on is corroded?
Here's a picture of the case.
If it won't move with a little persuasion I'll pull that plate off the left side of the picture and see what it looks like. I'm sure the gear lube could use a changing anyway.

 

I couldn't get my front wheels to engage in 4 low the first time I tried shifting since I got the truck. Dana 24 in mine as well. I believe it was the rod on the driver side that was not moving. After working it with a prybar and a hammer it came free and works fine now. I also took the opportunity to adjust the rod so the shift lever didn't have to hit the dash in 4 low. Good luck!

 

After working with hammer taps and a squirt oil can and pry bar I can get it in neutral now, and will work a little more to get to the other two positions. Thanks for letting me know I'm headed in the right direction.