The AAV only matters to a team, if they are going over the tax line. It doesn't help or hurt, otherwise.
Hosmer signed a $144M/8 year contract that comes to $18M on the tax budget for all 8 years. Nothing changes that, except maybe who it counts against, if a trade happens.
Yes, the Padres were paying him $21M for 5 years, and the tax line was only $18M, but it shows $18M for all 8 years, if you look at previous totals on cots.
When they traded Hosmer to the Sox, they agreed to pay all but the min salary for the remaining contract. That means they pay the Sox $13M a year minus the min salary, but on the tax line, it should be $18M minus the min salary, as that line never changes. The $18M counts on someone's tax line, and IMO it should be SD, since they are paying almost his whole salary.