The amount of tooth extraction infection would actually dictate whether you should go to another dentist or not. However, assuming that it is not an emergency situation, there are remedies that could bring you relief. Just like any wound, you would have to clean the area, disinfect the area, and then bandage the area. For future reference, always have back up doctors available to treat you for any ailments and second opinions.
The first task is to clean the area and for this, we will need to use some hydrogen peroxide. This is a very fast acting agent that functions like an acid. The reason that hydrogen peroxide should be used is, firstly, to get rid of any damaged and dead tissue in the area as well to clear out any infected fluids and pus. To apply hydrogen peroxide, first get something that you can bite on, as you might have to deal with a certain amount of pain. Open your mouth and under a bright light inspect the area for dark red, swollen gum tissue and any pus formation. Lean back and pour some hydrogen peroxide into your mouth. This part will hurt and you must leave the peroxide in your mouth for at least five minutes. Once this is over, rinse your mouth with an antiseptic mouthwash. This will also possibly sting so you need bear the pain for this as well. Finally, you need to cover the area and this can be done by placing some cotton wool packed tightly into the wound. Do this at night so that you do not end up re-infecting the wound by eating.
Since the area was infected, the chances that the gums were also inflamed are quite good, so you need to take herbs that will reduce infection and work as antibiotics. For this, add some ginger flakes to tea and drink it. The tea should be filled with the taste of ginger to the point of it being pungent and slightly difficult to drink. Next, since you would have to be on a liquid diet, have a lot of garlic that is mashed into a soup. Again, this should be a pungent preparation as the garlic is your natural antibiotic and must be taken in a dose that is strong enough. The most important part that you would be waiting for is pain relief. There is an herb for this as well, which is clove. You could swill some clove oil around your mouth whenever you experience pain, or you could chew on a dried clove through the day for continuous relief. Following these few suggestions would provide you with relief from your pain and would also help in reducing the infection.