In our opinion, you shouldn't have any problems mid to late May. If it does snow it won't last. Just allow plenty of time for extra night stays and watch the weather.
We rarely drove interstates with our 40' unless there was absolutely no other way to get from A to B. Many secondary roads are easier to travel. Even I-70 out of Denver has a very high elevation and that's where you might run into snow.
You might even consider taking Hwy 50 across Colorado, depending where you want to go. You could take it to Montrose and then north to Grand Junction. It's a very good highway. We've traveled on it often. The only issue on Hwy 50 would be Monarch Pass between Salida and Poncha Springs, CO. The westbound descent is 6% for 9 miles with some 30, 35 and 45 mph curves - not serious switchbacks. There are no runaway ramps.
Compare that to I-70 westbound at Vail Pass where you'll have a 7% descent for about 10 miles with with 2 runaway ramps.
You should have absolutely no issues in Moab late May. In fact, it could be very hot.
Have a great trip! It's a favorite area of ours.
You also might want to purchase the 'Mountain Directory for Truckers and RVers' Western Edition. Google and purchase it directly from the editor. Amazon's price is extremely high for some reason. It's been very useful for us as we enjoy driving in the mountains.