130munch

So, I thought I was being smart by getting a new high efficiency Kohler toilet, but after only 1 yr it started running non-stop, while not filling the tank enough to flush.

I thought I'd replace the inner pieces, which I'd done on my older toilet before, but much to my surprise none of home improvment stores carry the newer inner flush kits - they said I had to order direct from Kohler.

Before I go that route does anybody know of any common things I can check & try to fix on my own with these new high efficiency toilets, before ordering new internal parts?

Redwood

I'm guessing you have something similar to the parts in this link.

Kohler Class 6 Parts Diagram

Congratulations on owning a Kohler.
They often build toilets using proprietary parts and as you have found they are often difficult to find. Unless your local Kohler dealer stocks the parts you will be ordering.

Check the curtain valve to make sure the chain is not binding and allows the curtain to fall in place properly. If it is falling in place properly you probably need the #GP1059291 seal kit.

If you do have to order the seal call 1-800-4Kohler for parts. You may have warranty coverage as well but if you can get a $5.30 part local I would.