I like the idea of a primary DH, rather than the (IMO) foolish option of using the position to rest players. The position is basically a freebie for getting a productive bat into the lineup without worrying about defense.
Internally, the Sox are somewhat limited unless one prefers a Dalbec/Casas type at the position. To me, not the ideal solution.
If the Sox cannot keep either JD or Schwarber, then the other options on the market that might not break the bank do include Mark Canha, Joc Pederson, Jorge Soler, and maybe Eddie Rosario. The dirt cheap longshot is Renato Nunez, who showed once that he can be a slugger, but primarily just struggles to stay in MLB. But his spray charts show that Fenway might actually agree with his dead pull power. If he is left out alone with no dance partner again, absolutely get him a MiLB deal at minimum...