@DavidPostill gives some great insight into the utilities side of things and what to do if you think she's actually at risk of physical harm. As you mentioned though, you're hesitant to call SS without a clearer reason, so I'm going to run with the assumption you won't do that.
In all honesty you've already gone above and beyond to help a stranger. I doubt, based on the details provided, that you phrasing the question a little differently would have changed her response. Let's briefly break it down by level of involvement.
- You overheard a stranger's conversation and decided to interject. Already that's pretty forward, but your motives are good so it's not like you really went wrong there. Some people might be offended by that, but it sounds like this lady was already willing to discuss the situation with strangers.
- You offered to check with the utility company for her. This goes from a casual interaction straight to something much more personal. You're not offering to help fix a flat tire, you're getting into someone's personal life and (as you noted) could even impact their finances inadvertently. This is even more forward, considering you've only known this person for a few minutes. This seems to make you uncomfortable, and it should. Getting into finances of a friend is already murky water, let alone a stranger.
- You're now wondering if you should have taken it even further by physically involving yourself in the fix. In most situations this opens you up to being responsible if something goes wrong (you break something, are injured, she's injured, etc). This is almost always a bad idea, if just for the potential legal complications.
It's great that you're concerned and want to help, but you also need to be aware of how much you're entangling yourself in a stranger's life by trying to fix their problem. Interpersonal relations typically follow a progression as two people get to know each other. It starts with non-personal talk (weather, sports, etc), then moves to personal details (likes and dislikes, work, etc), and only after that would it involve any mutual arrangements out in public (lunch, coffee, etc). The next level above that is sharing of personal space (i.e., inviting someone into your home). It's yet another level above that to trust someone with decision-making, like fixing their electrical problem. You tried to jump from complete stranger all the way up to the trusted decision-maker in only a matter of minutes. Even if you were an expert electrician, that would still be a very pushy/pretentious thing to do.
I know that sounds like pretty strong wording, but I'm just pointing it out to set up for this...
The bottom line:
You've done everything reasonable to help and have been refused. Unless there's clear and present danger to warrant forcing the situation, it's not your place to do so. You've done all you can, and you shouldn't hold yourself responsible for her situation.