Pull the coax from the rig, use an ohm meter to check for a short between the outer shield, and the center conductor. try it without the antenna installed on the NMO mount.
the meter should read an open.
Next check to see if the surface that the NMO is mounted to is actually ground.
I had a ford ranger that needed every damn piece of sheet metal bonded to ground, they must glue them together after they coat them, because nothing seemed to be tied to ground.
Ideally, the outer shield, and the ground of the vehicle should be a solid short on your ohm meter. in other word, the outer shield should be ground.
I like to ground the chassis of the radio, and run 12v fused wiring directly to the car battery.
What kind of car is it?
mr_camera_man's idea of trying a different antenna is a good one.
Have you used the radio elseware with good results?
I have owned a ton of YAESU's over the years, with great results, but the last 2m/440 mobile was a lemon that got returned to the dealer.
It was a piece of crap.
Also, if you have another radio, even a handheld, try it on your new car antenna and see how it works.
If the coax tests OK, see if you can beg/borrow/steal another dual band NMO antenna for a quick test. mount it on the existing NMO mount.
That will help you isolate the trouble.
If You can borrow a VHF/UHF SWR meter, that would be a big help. if you've got high SWR, you can damage the rig.
Good luck.