Glacier bay sent new hot and cold cartridges but that did not fix the problem. Do all high arch faucets continue to run a little after truning off because the water is in the spout?

I have worked in The Home Depot's Plumbing department for the past few months, and from my experience there I have never heard of Glacier Bay, Delta, Meun, nor American Standard faucets act in that way. Of course, gravity is going to empty that arc quickly.. Maybe a drip or two if working correctly.
The problem sounds like it stems between the "motor" and the "casing" of the unit.
If you bought it within the year and still have your box, bring it up to your local Home Depot and we'll do our best to get you a new functional faucet.
D

Depends on you definition of "run a little " Try catching the water in a glass immediately after shutting off the handle. If you catch about a 1/4 cupof water , give or take, it is normal. There is at least 1/2 cup of water in the stem above the valve, that part of the stem making the high arc. When you shut off the valve that water is still in the arc, water past the high point of the arc will run out and likely siphon an equal amount out of the stem