Yeah, it's easier not to switch tires. I have extra road tires and I'm too lazy to switch.
I road train w/ Ritchey Speedmaxs ($30 for 2 on ebay). I can get them up to ~90 PSI on the road. On my commute training rides, I have a mix of paved and gravel. I also use these tires for singletrack as well. Probably won't cut it in muddy conditions since it's only a 30c, but for the mix of road and trail, works for me.
There are others that'll work too, like Michelin Jets, Vredestein Campos, etc. You'll want tires w/ high PSI for use on the road. When you take them on the singletrack, just release some air so you don't pinch flat.
CX season is next month, you have some time to practice, catch a clinic or two. Don't know what the CX calendar looks like in AZ. From my understanding, some races have a beginner race (no prizes) that should be good for a newbie. I'm new to this as well, and hopefully I'll have the skills for Cat C too.