Someone in this group suggested varnish as a "sealer"
I removed the loose tiles, ( yes, the floor was clean ) and used spray varnish on the vinyl. ( it's what I had on hand )
Re-installed the stick-downs, and, after several weeks, it looks like those tiles aren't going anywhere.
Thanks to whoever posted the tip. I'd definitely suggest a coat of varnish over the old floor before installing stick-down tiles. It worked for me !!