Ok, here is my problem.

When I turn on the water in the bathtub and then engage the shower, water continues to flow out of the shower head and the bathtub spout at the same time—I have the gate diverter pulled all the way up.

The water comes out of the spout very rapidly, too.

Could the diverter be faulty?

Thanks, guys.

Hi jakester.
Yes, your tub spout diverter is at fault. Some can be rebuilt, but its usually best to just replace the complete tub spout. Do you know if your spout is a threaded spout, or a slip on spout? IF you can get the old spout off, take it to a plumbing supply store, or any store that carries plumbing supplys and have them sell you the correct spout.

I agree, spouts are easy and cheap to replace. If you aren't sure which type you have look under the spout with a mirror, see a small hole it's a slip on, no hole its threaded. Slip on has a set screw and you will need to see it well to know what to use, some are hex set screws and others are standard screws

Just so you know the difference here's both spouts. Good luck, Tom