Do I need a new battery?

Including an isolator so you cannot flatten the second as well.
It is certainly an option - though I would want a diode circuit in there so that it always has a charging connection.
On the whole though - I'd rather carry a Lithium booster.
 
I have over four years on my battery now with no trouble.

I do put it on the tender when it’s going to get pretty cold, or when I know I won’t be able to ride for a while.