Unlikely a cable adjustment. More likely an indent on the gear shift arm housing. I would first check your transmission and see how it is attached. Some have you moving a lever that has an detent (sometimes a hole where a dimple in the armature sends a feed back to the gear shift). This may wear down over the years or the linkage may become loose with wear and the dimple is no longer fitting into the hole.
If not on the engine it may be in your binnacle housing.
Good luck Sean.
I agree with John, in addition :
Sticky cable can also do this.. results in overshooting the detent in both directions.. Ya can check by removing the cable connection at the transmission and working the transmission lever by hand.. If the detent feels good, then with the cable still detached, move the control lever and see how much resistance there is.. Sometimes a stiff cable can be lubricated, and sometimes ya have to change it.. (PITA)
Do you mean it throws you into forward or reverse? That is hard on the transmission. Not to say if you attempt the maneuver in a tight environ you might be meeting new neighbors in a compromising manner.
For me it is probably just a learning curve. I'm pretty good now, but last week I was teaching my wife and she was missing neutral, it seems to sit well enough in there and I've so far had no issues underway for several hours of motoring. We aren't very high stress on the lake so that probably helps.
I manually operated the lever (see pic) on the transmission with the cable attached. It moves smoothly but has approximately 3/8” of play in neutral. I’m thinking it’s the lever detent?
Where is the lever detent located?
I did not go inside the binnacle yet...
Lots of good advice here about checking on the transmission shifting end. Also, remember that your throttle and shift cables are "consumables" and do have to be replaced every so often. The internal cable structure and original lubricant will fade away with use. They do have a relatively long life, but it is finite. Getting 20 years out of a set is not unusual, but past that you are into the 'twilight zone' of maintenance.
We replaced both about 10 years ago (1988 boat) -- immediate difference in how slick the movement was, and yet would still hold position without slipping. Still good today.
If the cables were installed "by the book", they'll be secured every couple feet with cable clamps or zip ties. That's the PITA. If they're just routed and not tied down down, replacing should be easy. Disconnect both ends, tape the new to the old, and pull it through.