A.D.
i will be disagreed with, but as a single mom i know this to be true. if your child is absolutely inconsolable, it does not hurt them to be put in a safe place (her crib, her infant carrier, her playpen, etc) and left alone for 5 minutes. sometimes, that's all you need to restore your sanity and collect yourself. as long as she is safe and not a danger to herself, it is not going to send her into therapy to give yourself a mommy time out, nor will it damage your relationship with her. my daughter is nearly 4 and i had to give myself plenty of mommy time outs when she was younger. i can safely say that she is the most secure child i know, and even her pediatrician told us he rarely sees a mother daughter pair as bonded as we are.
as for a long term solution, have you considered having someone from your support team (friend, neighbor, family member) come over to relieve you for an hour or two every couple of days? i'm sure people will be happy to help you out. or, is it an option to call your ex and say, hey, i need some help with your daughter, can you spend some time with her so i can take a bath/do some shopping/take a nap/etc? if those aren't options, contact people you trust for babysitter recommendations. you'll want to get to know the babysitter first, see how s/he interacts with your baby, make sure they can be respectful of your wishes, but a regular babysitter can enrich your child's life and help you out immensely. i'm actually still in contact with the family i spent my teenage years babysitting--the oldest is actually one of my co workers now! but when you're a single mom and you don't have a strong support system, a regular babysitter is great peace of mind, as well as a life saver.
i know it is so hard to see your little one in pain! teething isn't easy, but just remember that it doesn't last forever, i promise you!