Class B job with class A license?

If you want the job, go for it. Just keep the Class-A cdl up to date in case you decide to make a move or the company puts you in a Class-8 truck later, which they probably will.
Look at the positive side; the day your hired, you're already senior to those that haven't been hired yet.

 

I would ask them to at least road test you in a class a rig so at least you are qualified to run both at that company.

If you just got out of truck school go look into an ltl carrier, then you can do p&d, be home nightly with weekends off and still have class a experience. Most ltl places will give you a shot doing city work or do a dock to driver program. Old Dominion would take you as well as con way with no experience.

 

Yes Old D, Con Way, UPSF, Saia, Estes, Pitt. And a bunch of others. Just Google ltl carriers they all have terminals. Stop by and ask to talk with the terminal manager at the shop. The T.M. does the hiring/interview.

 

I like your avatar; I used to drive for USF Logistics. It was non-union and I pulled a reefer coast to coast. Did about $55K per year there.

 

Yes Reddaway isn't bad I'm non union at our shop and I just did $1,800 this last week. Not bad with having full medical paid for.

Bring in a Resume and just explain your situation and pick their brain. Applications are done online. If you want, go online first and apply for open positions then swing by the terminal for places you applied for.