He's going to want more than $300m overall, and likely wants to be the highest paid position player in AAV, so I'll go with 9/350. If I had to guess a team, I'd say the Giants. I don't think the current Red Sox FO is going to sign a contract like that, and Cohen's going to have to replace practically his entire rotation next season (deGrom in particular), as well as dealing with FA for Diaz and Nimmo, so there's going to be a limit to what he can spend. Dodgers don't need him. Angels are trash and need to re-sign Ohtani, plus they might be sold, so I'd rule them out. I don't think Cashman wants to go that high, but if Judge breaks the non-roid HR record and wins the triple crown, then there's a good chance Cashman/Hal get run down by an angry mob if Judge leaves.
From a branding standpoint, I think Judge would be foolish to leave the Yankees. He is much more marketable as a Yankee than as anything else, but that may not be important to him. I'd say it's going to be the Giants or Yankees, but I'd lean towards the Giants.