Where did the 5th go?
The county sheriff is alas so full of it that it is overflowing through his ears, taking the remaining two of what passed for him as brain cells with it.
In keeping with what you mentioned in your post, arrests (actual and physical or issuance of citation in lieu of the promise to appear) are made in one of three ways; (1) when crime is committed in the presence of the peace officer or (2) crime is witnessed by a private person only and so a citizen's arrest will have to be made and (3) pursuant to a warrant issued by a judge or a magistrate. Unless he is talking on the strength of one of the above three, he had better hold his peace (no pun intended) and take a hike as the rest is nothing but hot air. So, in the obvious absence of (1) and (2), the sheriff will have to proceed by way of option three and I would give almost anything to be in the courtroom where he will be attempting to persuade a judge to issue an arrest warrant on suspicion of garbage dumping.
And it is also painfully obvious that the sheriff did not have the presence of mind to at least run his brilliant powers of deduction by someone who knows a little about the criminal law and procedure and the U.S. Constitution; ohhhh, I don't know, like the City Attorney or the District Attorney before opening his mouth and threatening your husband with charging him with an "extension of the first offense!" Otherwise he would have been told that a person cannot be prosecuted for the same offense twice as it would be a violation of his Fifth Amendment rights and protection against that universally known little thing called Double Jeopardy.
Sounds like this guy has a garbage dumping complex brought on may be by childhood memories of being tormented in his capacity as the school's dumpster monitor. But whether he has P.T.S.S. or just talks B.S., he has no cause against your husband and had better cease bugging you.
Not to worry!
fredrikklaw