Does the buyer need to pay the LLC or can the buyer write the check out to me personally so that I can go ahead and close out my LLC books?
Ideally, the check should be paid to the LLC, since the entity was the legal seller of the business.On the other hand, if you dissolve the LLC, you would be the successor in interest of the LLC's rights, including its right to the monthly payments (assuming you are the LLC's sole member).
Typically, it would be prudent to maintain the LLC until the buyer has completed payment.
In this case, the buyer could pay you directly.
However, the advantage of maintaining the LLC is that it acts as an extra layer of protection in case some liability arises from the business, whether due to the buyer or third parties. Typically, it would be prudent to maintain the LLC until the buyer has completed payment.
Granted, dissolving the LLC would simplify your tax filings and the like. However, your legal exposure to any blowback from the business would increase. Hence, deciding between these alternatives comes down to your risk tolerance and judgment on the likelihood of liability from the business.