I don't know much about boats, but the first thing I'd look at is the linkage between the shift lever and the transmission. See if anything's bent, broken, out of place, jammed, etc.
cable may be stuck or broken / if hydrolic, check level and fill as, if, needed
Looks to me like you just need to pull the levers apart and soak 'em in diesel oil, just to get past the rust. Then work 'em back and forth till they are moving easy. Use enough red waterproof grease on the joints and cable clamps to keep it from corroding.
The cable is probably toast, but it ain't hard to replace that. Old steel control cable had a black ABS sheath, a coil spring inside that, and a push-pull cable inside that. The longer the control line the thicker it has to be.
Swap that out and put on all new hardware. Then after that you only have 999,999 other things to make you want to sell that boat.
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