Yes, ideally it should be dry but a little oil on it will not hurt.
That is called a shift shaft seal. Replace it.
get the new seal from your local shop or online Remove the shift linkage/shift lever, use a pick or small flat head driver to pop out the old seal, clean the shifter shaft and case opening. oil/grease the new seal and pop it in with the open side faceing the motor
try flushing the transmission and putting a new trans filter in as well
Check oil level, and any leaks on output shaft for seal or baring damage - and if gear select linkage is visable then check for strange ware or hiting cables or body
Brake wheel cylinder or axle sealBrake wheel cylinder or axle seal
Remove the dust cap on the brake drum. There is a 30MM laminate nut under it. This is torqued to 192 Ft lbs remove it and the brake drum will come off leaving the wheel spindle and brakes in place. The drum actually hold the bearing a new seal should be added as well since you have the drum off
Remove the right rear wheel/tire. Remove the large nut and cotter key, than it is best to utilize a gear puller for this next step, if you utilize a pry bar be very careful not to scratch or damage the inner metal hub that mates to the hub seal. The hub will come off, next remove the small bolts holding the black drum cover in place. Once the outer cover is removed than the brake drum containing the wheel studs will come off. You may have to back off on your brake adjustment to get the drum off, this allows enough clearance between the shoes and drum for removal. The main components to replace are the shoes if worn, the large o-ring seal for the drum cover, and the main hub/drum cover seal. These parts are available aftermarket a little cheaper than OEM. Try Cheapcycle parts.com, they have very good prices. Good luck, Jim
my 1990 tracker has that exact same problem, what it is a a problem Geo has with its seals, what causes it is a the seal around first gear shrinks and keeps it from shifting and the cold makes it worse what you need is to have the transmission rebuilt, but if you cant afford that i would go to your local auto parts store and buy some transmission swell and seal. what it is is and additive that you add to your transmission fluid and it softens and swells the seal for better sealing.
It could, if they have the same gear box.
you can try wynns gear leak stop-it thickens oil and swells the seals.
You need to replace an oil seal on the gearbox.
A blown out rear axle seal will cause grease to leak out into the brake drums