Other sports have precontract agreements. For example a soccer player in the last six months of their contract in England can sign and agree a contract with a team in Germany for example. That way it secures that player. It's not employment by two teams it's just they go from one team to the next when their current contract expires.
Granted the NFL adopting such a method could cause issues like with expiring contracts signing with other teams, but if they just let the head coach candidates sign that precontract it secures them and prevents people backing out without that team's consent and agreement, and in turn a team like the Colts would get compensated for the Patriots last minute power play to keep their coach after he had given his word to the Colts that he was joining them.