A Totten Trust is simply a pay-on-death account which avoids probate for the amounts in the account. When the person who created the account dies, the amounts still in the account at that time automatically pass to the person named as the account (trust) beneficiary. This means that the funds do not first have to pass through probate.
However, the creation of a Totten Trust would have no effect on your potential Medicaid eligibility. That's because while you're alive, you have access to all the funds in the Totten Trust account. So, Medicaid will count the amount in the trust account as part of your total assets. If the amount of these total assets is more than allowed by Medicaid in your state, you'd have to use up most of these funds -- down to the state's Medicaid asset limit -- before becoming eligible for Medicaid coverage.
The only way a trust can remove assets from Medicaid consideration is if the trust is irrevocable, the person applying for Medicaid coverage no longer has any control over those funds during his lifetime, and the person creating the trust is not a trust beneficiary. To learn more about trusts that might permit Medicaid eligibility, you can look at the Centers for Medicare & Medicaid Services Web page Treatment of Trusts