More than likely, water has seeped in around the gear shift lever and has the transmission partially filled with water. With age, the shifter boot gets cracked, and lets water in. It sounds like there's enough in there to be up on the ring gear, and not allowing the differential to turn. Hopefully you have a warm shop, and be able to get it in there. At some point, drop the diffy plug, with a large pan (something like 10 gallons) underneath and let it drain.
Here is a link to more than several manuals for a 530 Case. The 430 and 530 are close enough, they share many similarities.
Case 53 CK - Manuals This will get you by, until you find the proper manuals for it.
When you get the serial number, you can go to the Case/IH online parts catalogs, and select the model that fall within the serial numbers listed.
Search 43 | MyCNHi US Store to familiarize yourself with it, and its function groups
Enlarging the picture, I can see it does have the manual shift reverser/shuttle shift. Word of warning, don't haunch on that lever very hard, and make sure you are completely stopped, before changing directions. The collar that surrounds the shaft, that shifts from forward to reverse, is made out of a cast, pot metal type material, and wear very easily. And, there is a very small spring/roll pin (like 1/16th" 3/32") that holds it on the shaft, that either the hole gets elongated in the softer collar, or will shear. If there is some slop in that shift lever, then one of the above has happened. You'll need to pull that shifter's cover, to see what I mean. Wear points will be evident, once you look at it.
On my 310B, it had been haunched on enough, they broke the shift shaft, and someone did a poor job of welding it back together. I ended up getting another used one that had some wear, but not broken. I drilled the hole out larger, and put a big enough roll pin in it, I could tap a smaller one inside it. Luckily, I found 2 on ebay for $35 ea. Got the second one for a spare, just in case, and let it go with it, when I sold it. But the first one held up for the 2 years I owned it. When as it should be, shifting will be somewhat crisp, and no slop when in either forward, or reverse. Again, just be at a complete stop before changing directions. The #2 reason for having good brakes, to be able to do this.
For as tough as those tractors are, that shift linkage, and the brake seals are the weakest points I could see in that style tractor. As far as the engine, and rest of the drive train, they seem to be near bullet proof, if you operate it responsibly.
*
late on instrument panel