I have a Winchell push mower (M5564S) Trying to figure out how old it is, and what it's worth.

Not too many people collect push mowers. Try EBay.

This is what I found, but tracing old links would be easier with a picture, or two.

The Story of Those Who Know Power Lawn Mowers | Agriculture Age | Preserving the Innovation of Agricultural History

https://www.smokstak.com/forum/showthread.php?t=72601

And this one could give you some interesting leads,

https://www.smokstak.com/forum/showthread.php?t=52637

Or these guys if they are still around.

FLOYD LORENZ AUCTION | Thornhill Real Estate & Auction Company