You should never have gear oil leaking externally. It's a totally sealed system.
There are three main seals to look at, the prop shaft, driveshaft (under the water pump), and shift shaft (where the shifter rod goes into the gearbox). In your case I'd put money on the shift shaft or driveshaft seal being the culprit.
There are also seals on the bearing carrier around the prop shaft, but they don't often leak.
The typical repair process consists of pulling the lower unit, draining it, setting it up in a vice and then pressure testing it through the fill/drain plug with a Mityvac style pressure/vacuum tester. Identify the leaking seals with soapy water, replace, test again. Some models will require disassembly to replace the shift shaft seals, but the prop and driveshaft seals can usually be pulled and replaced without.