Hi.
Last year, a friend and I did a century on the Katy Trail. We rode road bikes.
We both picked up a cheap pair of 28mm tires from Nashbar - (Hutchinson Flash, $5). They were okay. We went during October, and it had been fairly dry, so the trail was pretty hard.
I didn't find the ride quality or the traction to be a problem with 28mm road tires, but I woudn't consider their performance ideal in either regard. I certainly didn't have a problem with high friction or the tires sinking into the trail - but, like I mentioned, the trail was dry and hard. I think the perfect tire for the Katy would probably be something like a 32mm Panaracer Pasela or a 1.5" MTB slick.
Since getting a new bike for the Katy is probably out of the question, you should get the widest tires that will fit your frame and run with that. If you have a local bike shop, you might be able to find the ideal width tire more quickly... but the LBS will probably charge a bit more than Nashbar or Performance.
We had two mechanical problems on our ride: I flatted the rear tire and my friend broke a spoke. I think my flat was due to hitting a rut at speed with too low of a tire pressure. The Katy has, in places, a lot of deep ruts that run across the trail. They are often hidden by leaves, and hitting them at speed can be trouble! If you get 28mm tires, you will probably need to run 80-100 psi or so to avoid pinch-flatting on the ruts.
I read a tale of a Katy adventure where rain had caused the trail to get soft and the adventurer DID have problems with the tires sinking in. So hope for some dry weather soon!