Hrm.
Well, you could start supporting more RP benefits to get your foot in the door. It probably won't solve all of your problems, but if you start awarding players for RP with ingame benefits, such as better bonuses on die rolls, better resolutions, and maybe more exp, that could engage them.
For some more ideas, try grabbing the virtue/vice system from NWOD. I don't remember if 5e has an action die system, but if you give them a pool of rerolls or bonus die, then allow them to refill this pool by 1 every day rested, fully when a character indulges in a virtue, and by 1 every time they indulge in a vice, it could support roleplay. If 5e doesn't have an action die system, look around or think of your own.
Additionally, the Exalted ruleset has something called the Stunt system. Better descriptions represent better bonuses:
"At the lowest level, one [level] stunts require a good description of an action, adjudicated by the storyteller."
"Two [level] stunts require that the character interact with the environment in some notable action, taking advantage of the scenery that the storyteller provided. This can be physical or things the character knows about the world.. No detail may be contradicted, but minor details may be reveal in the context of the character's actions."
"Three-[bonus] stunts are singular acts of greatness, stunning bravado, and visual poetry, defined by their capacity to leave other players astonished. If there's any doubt that a stunt is level 3, then it's not a level 3 stunt."
Every level could be +2 to the die roll.
This could get your players to start doing the very basics of roleplay - emphasizing awesome situations and making the crudest of interests.
What do your character's backstories look like? How much have you used them?