Gas gauge dies when on city streets, but is fine on highways

[eskamobob1]

Hey Guys,

Pretty sure this is a nice simple one, but want to ask to make sure I'm not missing something. My gas pegs itself at the dead bottom (literally below where it goes when I have run out of gas) while I tootle around town and sit at lights, but 5 minutes on the freeway and its up and running like normal (4 trips in a row). My generator light also does not come on while sitting at lights. I am pretty sure this just means my battery is shot (the car has died probably a dozen times since it was installed due to infrequent driving at times and no tender). Anything else I should look into/check before I just replace the battery?

Thanks for the help in advance,
-esk