I am not licensed in TN, so this is a general response only. It is not intended, nor should it be construed as legal advice.
Further clarification is needed is to the identity of "they" in the second sentence of your question.
There exists a form of life insurance called "decreasing term". In general, one buys it when life circumstances are such that as years go by, there will be less need for the proceeds, and therefore, the death benefit reduces each year. That said, your comment that, according to the nursing home, were she to cancel the policy, it would get the money, does not make sense, because term insurance by its nature, does not accumulate cash value. That is, when premiums stop, coverage stops within a short time thereafter. In contrast, with whole life coverage, the accumulated cash value can carry the policy for some period of time depending upon how long premiums had been paid.
It is hard to give you a more definitive answer than the foregoing without seeing the policy. You should consult with a licensed life and health agent to review it and explain to you what is happening. You also need to determine if your aunt pledged the policy as security for her nursing home expenses (or is otherwise in debt to the nursing home)--that may be the reason for the comment that the the nursing home would get the proceeds. From the wording of your question, though, I am not sure who/what made that statement.