If the starter solenoid is hanging up mechanically, it will tend to draw excess current. The starter itself could be drawing too much current if its bearings are going bad (dragging).
(You know not to substitute a larger fuse. I once repaired a stereo receiver whose owners had responded to a frequently blowing 3.2A fuse by replacing it with a 30A fuse. The unit's $40 (1970 prices) power transformer became the fuse.

)