If we decide to assume guardianship of our incompetent uncle, will we be in charge of paying his bills?

You are not responsible for paying his bills with your own money. As guardian, you'd be responsible for paying his bills with his money, and if there isn't enough money, to either arrange more income (medicaid, for example) or to reduce his expenses. Having an attorney appointed as guardian is certainly easier than doing it yourself but it's very expensive. Unless your uncle has a large income and/or significant assets, have a probate attorney help you apply to be the guardian. As a guardian, you can't write a will for your ward or make gifts of his property. If he's competent enough to make a will (a person can be competent to make a will but not competent enough to make medical or financial decisions), you can arrange for him to make a will.

If you are applying for a guardianship of the person and estate, you will be in charge of paying his bills but with the assets in your uncle's estate, not your own personal assets. You can become the guardian but it would be beneficial for you to work with a probate attorney to consult with you as well set up the guardianship for you and advise you on the responsibilities of a guardian.