Dennis:
Are we talking about a light on top of the housing, or a light near where the power plug plugs in?
All of my battery compartments, and my B27, have (near where the power plug plugs in) a red light that illuminates when charging is still happening, and a green light that illuminates when charging is done. If yours doesn't, I would guess that there have been changes since my equipment was new.
The newer version of the battery compartment I have also has the flashing light, but it signifies that the flash is turned on, not that it is charged (it is switch activated).
The "dry fit" battery is the right type, and as I understand it the 60-38 model designation is correct. In the past, there have been other sources for the same battery (it comes from a battery manufacturer and is used in other equipment) and some of those alternate batteries don't have the "nose" on the side of the battery that ensures proper orientation.
A couple of other points.
My 60 CT1/2 battery housings have a moveable switch in the compartment where the battery sits. It is above the battery, toward the right side. It has three positions: "nc" - for nicad batteries, "dry" for dry fit batteries, and "pb" for lead acid batteries. I wouldn't be the least bit surprised if the newer models don't have a "pb" setting any more. In any event, you should check to make sure that yours is set to "dry", otherwise the voltage is probably wrong.
Also, the contacts in the battery housing do heavy duty work. I regularly clean them, and the contacts on the top of the battery, using a pencil eraser.
Also be sure that the ridges on the bottom of the housing that the battery sits on aren't damaged or bent. The battery should make strong positive contact with the contacts - when you put the battery into the housing it should clearly be making contact.
As far as I know, one of the reasons you pay the big $ for the Metz 60 series flashes, is that they are designed to prevent over charging.
See if this helps. I'm willing to keep trying if it doesn't.
Matt