To falsely state that you owe a debt when you have paid it is defamatory. However, to sue for defamation in the US (and most other places) one must establish that the statements are false. The question says that:
The debt was paid to her late husband, there was never a signed contract or receipt: we made the deal with a handshake and settled the deal the same way with no paperwork.
That would make the falsity of the statement that the debt is still owed quite hard to prove. Unless there was a witness or some other proof it would depend on your word alone, and a jury might not accept that.
You would almost certainly need a lawyer to file and pursue a defamation suit. Some lawyers specialize in such cases. Such a lawyer could advise on the specifics of such a case, and whether it would be worth the costs and fees which would need to be paid to pursue such a suit. One generally needs to prove that one's reputation has suffered actual harm to collect significant damages, as well as proving falsity.
Proof of Falsity
In The Plaintiff's Burden in Defamation: Awareness and Falsity by Marc A. Franklin and Daniel J. Bussel (William & Mary Law Review Vol 25 Issue 5, June 1984 it is said that:
... the plaintiff must establish the existence of a disprovable
defamatory statement and must prove the falsity of that
statement with convincing clarity. (* p. 835*)
Now, a wide range of statements that cannot be proved either true or false necessarily are absolutely protected. (p. 855)
In Wilson v. Scripps-Howard Broadcasting Co. 642 F.2d 371 (6th Cir.), cert. dismissed, 454 U.S. 1130 (1981).. the United States Court of Appeals for the Sixth Circuit concluded that certain constitutional principles announced by the Supreme Court required that the burden of proof be placed on the plaintiff. (p. 857 This was a private plaintiff case)
Although constitutional principles appear to require this shift in the burden of proof, reason and justice also dictate that the plaintiff should bear the risk of nonpersuasion on the central issue of the statement's truth or falsity. (p. 859)
although the defendant may contend in his motion for summary judgment that he has shown the truth of the statement conclusively, the defendant actually is arguing that he has shown conclusively that the plaintiff cannot establish the required element of falsity. See, e.g., Meiners v. Moriarity, 563 F.2d 343 (7th
Cir. 1977). The defendant may have seized the initiative on the issue in order to avoid the trial, but this tactic does not affect the underlying conclusion that the risk of nonpersuasion is on the plaintiff. (p. 859)
Trial lawyers often have complained about
the unfairness of having to demonstrate that a derogatory statement is untrue.134 Recognizing the difficulty of proving a negative, Roman law placed the burden of proof on the party who could prove the affirmative. What Roman law and current trial lawyers fail to recognize, however, is that not all negatives are difficult to prove. A detailed defamatory statement should be readily discreditable. In Gertz (418 U.S. at 326.), for example, the plaintiff was able to establish that he was not a "member of the 'Marxist League for Industrial Democracy. Modern discovery practices have also obviated much of the earlier concern about proving a negative. Furthermore, courts long have required plaintiffs to prove negatives in many other areas of the law
In short, when the plaintiff fails to prove that the allegedly defamatory statement is actually false, even though the defendant is unable to prove it true, the defendant will prevail, and no damages will be awarded.
Now, as to the issue of the photograph. The image would be protected by copyright. If you took the photo, you would own the copyright. If some other person took the photo, that person would own the copyright unless they transferred the rights to you in writing.
Only the copyright owner or someone authorized by the owner can file a copyright lawsuit. In the US, the copyright must be registered before suit can be filed. There is a fee for registration.
The person sued might assert fair use of the image as a defense. this is always fact-based, there is no way to say for sure if a use is a fair use until a court makes a decision, but one can review the four fair-use factors in a given case.
The law on fair use is found in 17 USC 107 The four factors listed in the law are:
(1) the purpose and character of the use, including whether such use is of a commercial nature or is for nonprofit educational purposes;
(2) the nature of the copyrighted work;
(3) the amount and substantiality of the portion used in relation to the copyrighted work as a whole; and
(4) the effect of the use upon the potential market for or value of the copyrighted work.
On the first factor, the use is neither commercial nor educational. It is not transformative. This probably leans slightly against fair use.
On the second factor, the image is not a news report or factual work like a textbook, but neither is it highly creative like a work of fiction or a work of fine art. This is probably neutral with regard to fair use.
On the third factor, presumably the whole image or a large part of it is being used. This would lean against fair use.
On the fourth factor, there probably is no significant market for the original image, so no harm is being done to the market. And if there were any market, the sort of posting described in the question would not harm it. This factor would lean in favor of fair use.
With some factors leaning each way, it is very hard to predict what a court would do with a fair use claim
17 USC 504 provides for damages in the case of a successful suit for copyright infringement. There are two main categories of damages:
- "actual damages and any additional profits of the infringer". If the photo was not being sold or otherwise earning money for the copyright owner, and the infringer was not making any money either, there would be no significant damages under this provision.
- "statutory damages". These can be as low as $200, or as high as $150,000. But under 17 USC 412 statutory damages for a published work are not available at all unless the work was registered before the infringement took place, or within three months after the work was fist published. Posting an image to social media would constitute publication.
One could also send a DMCA takedown notice under 17 USC 512 to the web site displaying the image. This might work. Only the copyright holder or the holder's authorized agent can do this. If the image is physically posted in the store, this is not possible.
Cease and Desist Letter
One could send the person displaying the image and false information a cease and desist letter. This is, in effect, a threat to sue. It may cause the desired action, but it has not legal force if not followed up by an actual lawsuit.
None of the options for responding to the situation described in the question is guaranteed to work. All of them, except a takedown notice, would require upfront costs to be paid, and would as a practical matter demand that a lawyer with experience in the area be consulted.