Small Engine Troubleshooting Problems? Ask a Mechanic Now.
Welcome to JUST ANSWER small engines, let me try to help you !! I don't know it all, but maybe together we can get you a satisfactory answer!
This is a standard gear transmission, and if it is not going into certain gears, either the shifter is not moving the lever far enough, and the linkage needs adjusted, or the internal gears are damaged. See link above. With it having a broken shifter, I would remove the linkage to the transmission shift lever and move the shift lever manually and see if you can get ll the gears, then adjust the linkage to match the transmission lever.
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I see question is still open, did you find problem?