Most likely a mechanical problem with the switch. While I don't have a 2600, Atari likely used an auto-repeat with the switch to advance repeatedly while the switch is active.
You MIGHT be able to fix it yourself.
This comes down to:
- disassemble device, without losing any screws or breaking anything
- examine switch for identifying marks or model numbers
- type numbers into a search engine, and see if any matches come up
- if yes, buy the part from an electronics supplier
- if no, you may be able to hack in something else but it won't be as pretty as the original device, and you may break it trying to hack it, but oh well it doesn't work properly the way it is now, right?
- use soldering iron and "solder sucker" to remove solder from old switch, without burning yourself
- remove old switch from board, noting orientation
- install new switch in same orientation, or it'll work backwards or not at all
- solder switch to board with rosin core solder, try not to inhale fumes
- test device before reassembly, without touching any metal shields or screws to electronics and blowing it up
- if good, reassemble back together with the screws and covers stored away weeks ago