Somewhat. But why treat yourself? You still need dental care. There are no home remedies or cures for rotten teeth. No magic pills or rinses or mouthwashes. Toothaches need to be treated locally by a dentist, someone who has years of experience treating these conditions. The sooner you call the sooner you will feel better. Please don't delay, the pain may soon be unbearable.
See your doctor. This does sound like it could be an infection. You definitely should see your doctor without delay, as these kinds of infections can get out of hand and turn into very dangerous, life-threatening problems.
Tooth Abscess. You have a tooth abscess! You are right, it is infected. The lump on your jaw is the abscess finding a way out. Antibiotics will help but will not take care of the problem. You need emergency dental care asap. If the infection blows out under your tongue, you can develop a life threatening infection that will raise your tongue and closes your airway. Not common, but I've seen it twice in my life.
R/o abscess. Go to dentist. The lump may be an enlarged lymph node suggesting infection. Antibiotics may help but you will need to drain the abscess or treat the tooth somehow.
Temporarily. The lump under the jaw is an abscess or enlarged lymph node. Antibiotics will temporarily help, but the problem will recur unless you treat the cause. See your dentist to address the primary problem.