You might to check if there’s any dental schools in your area,they often offer services in a sliding scale basis or sometimes even free.
Some government agencies have programs to assist indigent individuals requiring dental services.
If all else fails you can always drop in on a dentist (toothaches are considered “emergency” dental cases, and you don’t have to schedule and wait for an appointment that may be a week or more away) and ask about a payment plan for a non-covered extraction, if cost is really objectionable. Extractions without complications generally shouldn’t cost too much, though. If he knows you’re going to pay today in cash, he might even be able to adjust his pricing a bit to accommodate the lack of third party insurance, too.
A root canal, implant or bridge, etc., that’s going to be more involved and costly, so you really might need a payment plan for that.
But the dental school idea is always a good one. In that case you might only have to pay for materials used (such as Novocaine, for example) and a very nominal charge for the service itself.
If it is a mild infection it should eventually pass. If the infection is caused by an opening in a tooth you have an problem that is likely to get worst. Consider a discount program like Carington, or going into a smaller town or a poor neighborhood. In the mean time swig salt water or better yet mouth wash with some germ/bacteria killing guarantee whilst you decide.