1250 has a different linkage setup than the narrow frames, but my 106 and 107 both actuate the throttle a lot more than that, even when shut off. Something definitely isn't right, I would follow the adjustment procedures posted above first.
If the governor itself is busted, unfortunately yes, you need to do a total rebuild, as the governor is the last piece removed and first piece installed when rebuilding.
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'70 107 with k301 engine swap
'71 106 with 38" deck
'70 147 R with factory replacement k321, 42" deck
'61 Original with 38" timed deck
'63 70 "pinkie"
1863 with 54" deck
'46 Farmall H, '50 Farmall Cub
105 x2 (parts)
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