An OTH is a "General" discharge. General under other than honorable conditions. There's also Bad Conduct Discharge (aka Big Chicken Dinner), General under honorable conditions, etc.
A Dishonorable discharge in this case isn't about the weapons qualification being "valid" or not, it's about the fact that a Dishonorable discharge carries the civilian meaning of a felony conviction. A Dishonorable discharge will preclude you from even getting a carry permit in every state that issues one.
It may also be a NICS disqual to buy a gun, but I don't know that for certain.
Your OTH is of no issue other than when it comes to qualifying for veteran's benefits (GI Bill, VA loans, etc).
P.S. I have no official capacity on any of this, but used to hang out a lot with the legal officer on my ship.