Does a seller carried note have to be assumable when I sell it as part of an estate

The note transfers to you by operation of law through the estate, the same mechanism that moved the deed. Tennessee intestate succession and probate law treat a promissory note as personal property of the decedent, so it passes to the heir or beneficiary without requiring the payor's consent. The buyer has no right to renegotiate simply because the holder changed, and the absence of due-on-sale or assumption language does not create one. That silence actually works in your favor here: there is nothing in the instrument that gives the buyer a trigger to call the note or demand new terms. Their obligation is to keep paying under the note as written, and your obligation is to honor it on the same terms your mother extended to them.

The thing worth confirming before you do anything else is whether the informal extension after the balloon was documented at all, because that is the number doing the most work in this situation. A handwritten note on a printed form is enforceable in Tennessee if it has the essential terms, but an oral or loosely documented extension changes the maturity date, and if the buyer ever disputes those 14 remaining payments, you need something that shows the extension was agreed and on what terms. I would get a real property or transactional attorney in Tennessee to review the original note, whatever extension documentation exists, and the deed of trust or mortgage instrument, because a general practice attorney cannot tell you with confidence whether the security instrument was recorded correctly and whether it actually attached to the property in a way that protects you if the buyer stops paying. What does the extension look like on paper, if anything?