Yup definitely check the shift adjustments.
It's strange it cuts out when it jumps out of gear. That might come right if the adjustment keeps it in gear. If the adjustment doesn't cure the jumping out problem and the gearbox needs work, then I'd try to figure what was causing the cutting out before I fixed the gears.
If, after adjusting the shift so it's moving fully into high, it still has he problem, it would be a good idea to take careful note of what the circumstances are causing it to jump out. There are some tests/observations to be made before stripping the gearbox.