Your understanding is correct. A 709 is only needed for the year of a gift over the annual exclusion amount (currently $15k) . If there is no future gift, no future 709 need be filed.
Be sure to keep the 709. It may be needed when your mother passes away. Also keep the address it is sent to. (That is asked for on a 706 if it is filed).
Note there is a difference between an estate tax and a inheritance tax. An estate taxes tax everything the decedent owned when they died. That's why they have to deal with gifts one way or another. Otherwise you could just give away everything the day before you die. An inheritance tax applies not to the giver but rather the recipient.
Because your mother lives in IL now, that is the state that matters for estate and gift tax. There is no IL gift tax form, but it appears that lifetime gifts (over the annual exclusion amount) will be added back to your mother's estate (if she still lives in IL when she dies).
Because you live in MN now that is that state that matters for inheritance tax. (Don't get hung up on "inheritance" it can apply to lifetime gifts). Since MN (and most states) don't have an inheritance tax, you don't need to worry about that.
I assume from the language of your post that your mother isn't married. It is worth noting that the annual exclusion limit is $15k per donor per recipient per year. So if she were married and you where married each parent could give each of you and your spouse $15k per year = $60k and no 709 would be required. (I would do each payment separately. Married donors can "gift-split" but a 709 would be required).
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