I had a 310B with the same problem. Open this link to Case/IH
Case IH | Schematic, Manuals, Specifications and Diagrams for shuttle transmission control cover and control lever kits | MyCNHi US Store and it shows an exploded view of the shifting apparatus under the cover. Note the one shifting fork on the left. If memory serves me correct, you need to raise the left side of the cover up first to get the fork to clear the transmission case. Seems it had to be about the midway point between forward & reverse to get it out or it will hit on the front of the transmission case.
If you look close zooming in on the exploded view of the reverser shift lever you'll see a #6 woodruff key and to the right of it a 3/16" X 1" roll pin. The woodruff key plus the roll pin is the connection to part G16663 (can't remember what you call them too early and not enough coffee).
On mine, the shift lever rod had been haunched on so many times it had broken and a PO back the line had it welded it in the keyway was a bit wallowed because of this and had some slop. But the roll pin in mine had half sheared which would limit travel in the reverse direction. Part G16663 (I'll call shifting finger) is made of like potmetal/cast aluminum material and the hole in it was a bit wallowed too, plus the roll pin being half sheared.
On mine I got the roll pin out and drilled that hole to the next size roll pin, it was large enough inside to drive a smaller roll pin through it to make it hopefully twice as strong. It did solve the problem for me and lasted the 2 more years I owned it. But when changing direction I made sure it was completely stopped avoiding any gear clash. The shuttle shifted very easy, so I just used my thumb and index finger to shift it, so as not to cram it.
Seeing how fragile that shifting mechanism is, I got another one that just happened to be on ebay at the time. I never had to use it as my repair lasted until I sold it and it went with the tractor. I did warn the new owner about being careful when shifting the reverser, told him I'd repaired it and why I had an extra shift cover going with it.
If you need any other parts for the tractor here is a link for the Case 300 series tractor from Case/IH:
Case IH | 300-SERIES - Case Wheel Tractors | MyCNHi US Store They pretty much took a 300 series utility tractor and put the optional HD front ends under them and maybe a few other things. That was 12 years ago and discovered back then the best way to find parts is to use the OEM part number. I used Google search engine and got the best overall results.
Mine had a Prestolite distributor so you might want to check yours. I did a tune up on mine and even back then the distributor cap was $26 through O'Reilly's, seems the points were around $12. I will say that engine started and ran very well. Just touch the starter and it was running.