Assuming the $7 million AAV is accurate, this leaves SJ with ~$65.89 million committed to 19 players next year, with Thornton, Hertl, Tierney, and DeMelo (and Tavares) still to sign. This is assuming that they let everyone else walk.
Thornton might take a very team friendly deal, but it's still hard to imagine him getting paid less than $5.5 million at the very minimum. Hertl will likely be making $5 million plus. Tierney should get $3.5 million or more, and let's say DeMelo gets $1.5 million. That alone takes them to $81.39 million.
Say they move their two biggest dump candidates in Martin and Boedker (combined $8.85 million), replacing one with a $900k guy. That takes the roster to 22 players and $73.44 million. You're still not cramming in Tavares if he wants $10 million plus (which he should absolutely get), even if the cap goes all the way up to $82 million. Who can they easily dump for cap space and replace cheaply after that?
They might be able to manage it somehow, but it sure doesn't look like it will be easy unless you make some pretty generous assumptions about what everyone else involved will end up being paid, or they do something shocking like walking away from Thornton. It may not be impossible, but I think this signing puts a pretty big dent in the likelihood that they sign Tavares, personally.