Re: how do i know that my lower shift cable is bad
An Alpha outdrive (what you have) uses "dog clutches" (big notches) to engage the gears. Shifting dog clutches under power destroys them. The lower shift cable is connected to a cam that trips a kill switch when you go from neutral to forward or neutral to reverse. The kill switch momentarily kills the engine so you don't destroy your dog clutches. A binding lower shift cable holds the kill switch down too long and the engine dies.
Doesn't sound like you checked the boat out too well before you bought it. There's lots of other things (leaking bellows, gimbal bearings, u-joints, drive seals, rotton stringers, rotton transoms, internally corroded exhaust manifolds and risers, etc) that could be expensive that you should learn about or get someone to check out before you put too much money in a 30 year old boat ....