Thank you to everyone for the comprehensive treatment of my question. It’s more than I expected!

I’d like some clarification from DC or anyone who might elaborate on DC’s explanation. I think DC suggests testing the sentence by simplifying it to ascertain whether the objective or subjective pronoun fits. His shortened examples work well: “Serious gardeners use organic fertilizer or “My wife and I use organic fertilizer.” However, what if the husband/subject of the original sentence refers to himself only?:
“Serious gardeners like me/myself/I always use organic fertilizer.”

DC says that “Serious gardeners” is subjective, so “like me/myself/I” must be, also. He says “me/myself” are not subjective, so they’re wrong. If I delete “me/myself” and simplify the sentence per his suggestion, then I get the following:

“Serious gardeners always use organic fertilizer” or
“I always use organic fertilizer” … therefore …
“Serious gardeners like I always use organic fertilizer.” (This sounds odd.)

What have I done wrong here? Is “like I” correct here?

Thanks,
Helen

Douglas, Steve, and Chris,
Thank you ever so much for the helpful responses. I could swear that the increasing popularity of migraine-inducing expressions such as the one Chris mentioned, "Please return this form to the secretary or myself" is responsible for my oblivion. "Theirselves" are at fault! :-)
Myself is signing off,
Helen :-)