I don't have the jab, and I renewed my DOT and FAA 2nd class medicals in February.
To the OP, and the CDL question. As you have found in this thread, you will get a different answer from every one you talk to. What I did is consider the potential risk I may be exposed to in the event of an accident. I have a military tractor M915 and M373 trailer. They are both registered historic. If I get in an accident and there is a serious injury, I could be found liable for operating a commercial vehicle (GCVW of over 26K lbs) without the proper license. In the eyes of the law, that's what it is.
I decided it would be best to get a class A CDL, so I have one thing less to worry about. I'm not going to argue if you need a CDL or not, and I don't care what anyone else does. All I'm doing here is explaining what I did and why.