I was not impressed with the Genius counter either but I guess some people have come out ahead with them. Probably depends on the problem.
Maybe the radio dock you were using did not know when your battery was charged and overcharged your battery.
I think a replacement battery is $7.00. Be careful getting the old one loose. I thought I was careful but still managed to cause bleeding on my LCD screen.
Every site I have read makes it sound like the glue comes loose with minimum force. Maybe that is true when the glue is new but on these older iPods I have messed with the glue is very hard to get loose.
IMHO a 1st Gen is easier to get apart then a 2nd gen but just be careful around the antenna area as there are delicate parts and also antenna parts that like to get lost and are not replaceable.