I feel like Tulip is conveying not just anger or stubbornness, but some actual hurt. When she and Jesse argue for the umpteenth time and he leaves her at the gas station, the way she says she's not leaving without him makes me think there's a deeper pain there that she can't alleviate on her own. Between that, the way she talks about Jesse in the traffic stop scene, and her flashback scenes (she looks so terrified and blindsided when the speeding car takes off without her), I'm incredibly curious as to what is motivating her to kill Carlos and what role Jesse has to play beyond the muscle or the wheelman.
The problem, I think, is that the writers are taking their sweet time to get us there because they want to explore Jesse's journey as he tries to save the town, in addition to all the other ongoing storylines they're trying to set up at the same time. That doesn't leave Tulip with a whole lot to do while she's waiting for Jesse to change his mind, not unless she's going to strike out on her own (which she's obviously not, otherwise she'd have taken a hint by now). It actually might have been better for the story to introduce her later in the season, or make her take longer to find Jesse, if they couldn't figure out what to do with her now, but I guess you can't wait that long to introduce your lead actress on a television show.