The first thing to check is the antenna cable boot. It may look like it is sitting down on the connector, but if it is riding up just a touch, water will run down the cable, around the boot and back in the bottom. There is a nut on the cable connector on the base that may well be loose. Pull back the cable boot far enough to remove the cable, then tighten the coax assembly on the base. After you do that, make sure that the boot is tight on the coax cable. Sometimes water will run down the cable and the boot isn't tight on the cable and it will leak past it. I have a small tie wrap holding mine as tightly as possible and it won't slip up on the cable at all.
Next, even though you say there is no water dripping on the center shaft, there is a rubber oring called the "quad ring" that seals the shaft. It rotates with use and can easily become brittle. There are instructions in the owner's manual to remove a small rubber plug and spray silicone spray into the hole to lubricate the gears, the shaft and that oring tow or three times a year. If it isn't kept lubricated, it will flake away and water will run down the shaft.
You say the antenna is "loose" on the roof. If you mean that the base moves or flexes with the OSB underlayment, that's normal. If you mean the base is moving and the OSB is staying stationary, then you may have either some screws that worked loose or even worse, a "long time leak" that has rotted away some of the OSB and the antenna is loose.
ADDED: I always think of more things after I post, but I was just thinking about your statement that the antenna is loose on the roof. If you mean the actual part that moves up and down and rotates, there is a nut on the inside that tightens the shaft to the antenna extension, That nut may be loose, it should be "fairly tight" but not so tight that the antenna will not rotate freely. Just thought about that, and your leak may well be right there.....
If you think about it, the base is sealed along the outside with DICOR, but there is nothing sealing the inside of the base (around the shaft) so any water that gets in can migrate to the wooden surface under the baseplate if it isn't tight and if the quad ring fails.
It's not difficult to disassemble the antenna, just pay attention to what goes where. I'd take it apart, check the OSB and reassemble with a new quad ring and reseal with new DICOR.
Here's the link to the owner's manual:
http://dodgenmobiletech.com/sites/de...%20Antenna.pdf
Good luck, and yell if you have any questions.