Well, welcome to the working world. I, too, have been accused of having "the curse of the competent." You get handed all the work there is, because you'll do it very well and do it on time.
I agree with all the others: find a new gig. Unlikely your company will value you. You could do the offer/negotiation thing - and it's not a bad play - but long term, my experience is that an employer like that usually won't work out.
With that said, I have a friend who did as they suggest and used an offer to leverage a better offer at his company, and then his career really took off, actually. So it's not an absolute.
But generally, you're better off leaving I think. And prepare to find the same thing again, too. Many employers will burn right through the most competent employees, or burn them out, especially if they don't start putting in boundaries, negotiating, and so on.
Another tip: be sure you know how to say "no." I struggle with it. But it's vital: you want to say "no" and protect your schedule, your home time, and so on, especially when everyone else is accomplishing very little. Be candid - and charitable - about others' work, but if you consistently allow your employer to run over you, many many managers will run over you, again and again. In fact, I've seen employees burn out really quickly that way: they refuse to say "no" until they pass out from exhaustion, then they either quit on the fly or are fired, which is a real shame.
And yes, all the more reason to start planning your escape...