Wobbly unlevel tank on Kohler Highline K-3427

I have a toilet problem... I discovered today that the K-4620 tank of my Kohler Highline K-3427 toilet (with "Ingenium" flushing system), installed in 2005, is rapidly becoming very unlevel and loose. There is about a 1/3-inch gap on the right side of the tank-to-bowl connection (much smaller on the left), and the tank wobbles far more than the 1992-era Mansfield 3.5 gpf toilets in the same house. There is no leakage in the Kohler that I can see; however, I have not run a leak test.

Also, this might be the reason why the flapper of the Kohler in question gets stuck every 2-3 weeks and I have to reset the flapper.

This brings me to my final point: Can you tighten a tank after installation of the toilet, or are you simply out of luck?

You can always tighten a tank up.

It has no effect on the flapper though.

That could be a stiff trip lever, or pehaps a chain adjustment.
2005 is getting up there for a flapper though, you may want to change that.