Jazz, teething babies are quite demanding. It's okay to let the housework go for a while and just tend to your little one and yourself. A little bit of dust won't kill anyone. Have you tried the baby orajel? It was a lifesaver when my kids were teething. Frozen washcloths work well too.
You are such a great mommy, being so concerned about your child's needs! And don't worry aobut your partner working too much to help out. Being a mom is a 24/7 job. It never ends. We don't get days off, holidays, or weekends. Give yourself some credit for doing such a wonderful job! I doubt your partner minds helping you out. Raising a baby is a lot easier when you make it a joint effort.
And really, with the housework, prioritize. See what REALLY needs done NOW, and let everything else go. Plus, when my kids were babies, I'd put them in the BabyBjorn carrier and just carry them around with me while I was doing housework. That way we got to bond, AND I got stuff done. My babies LOVED it! Do you have one of those? Have you tried that?
Good luck to you, being a new mom is so incredibly exhausting.

