Expedited vs. regular trucking

Last Summer I asked a driver who drove Load 1 whether I needed a Class A or B to drive the type of staight traight truck that was driving. He told me niether one because the truck was under 10,000lbs. As I mentioned a few weeks ago on this site, I have three preventables(two of which will hit their three year mark in july) on my DAC. With these two factors, do I stand a better chance at getting a job with an Expedited Carrier driving a straight truck(or what ever those funny looking trucks are called that look like a combination of a tracter w/ sleeper and a box truck). Thank you!