In most states, annuities are considered assets and the proceeds from the annuity would have to be spent down to the Medicaid asset limits before you or your wife would be eligible for benefits. Otherwise, anyone could shelter assets by investing them in an annuity.
However immediate annuities, those in an irrevocable payment mode, are counted as income rather than as assets because they can no longer be cashed out. The purchase of the immediate annuity or the transition into payment mode must meet certain requirements in order to not be considered a disqualifying transfer of assets. And, of course, increased income also can affect Medicaid eligibility in certain situations.
All of this said, the rules may be applied differently in your state. So, we recommend consulting with a local elder law attorney. To find one near you, go here: https://www.elderlawanswers.com/elder-law-attorneys.