This is going to be brief: the fact of the matter is, as far as "sense" is concerned, Marianne just doesn't have it. She's always getting herself into a tizzy about this or that, and her plot arc is directed purely by her emotions (until the very end, that is). She's something of a self-indulgent romantic, and she very rarely thinks about what's practical or proper.
OK, here we go. Marianne is all about sensibility – that is to say, she's all about passion. She doesn't really think before she acts, and she believes firmly that one should be directed by feelings, not logic. Marianne pours out all of her feelings into tempestuous sessions at the piano, and equally tempestuous rants. She hits both dizzying highs and terrifying lows over the course of this novel; the former come when she's head-over-heels in love with Willoughby, while the latter strike after Willoughby dumps her brutally. Marianne in love is a thing to behold – she's euphoric and radiant, and her happiness is infectious. When she's down, though, she's really down. Her depression and subsequent illness put a damper on everyone around her, and it seems that her emotions influence the lives of her friends and family almost as much as they influence Marianne herself.
Marianne's "sensibility" represents the kind of literary heroine common in the so-called "novel of sensibility," or sentimental novel (a prime example is Samuel Richardson's Pamela). These heroines were typically romantic, innocent young girls, inclined to swoon and be relatively confused. Austen's witty novel takes this sensitive heroine-figure and displaces her – instead of being in a novel that glorifies emotion and sentimentality, Marianne is stuck in the real world, a place of money, practical marriages, and – you guessed it, good old fashioned common sense.
In the end, impulsive Marianne comes to realize that her unleashed emotions almost killed her – literally (her near-fatal illness was the result of her impassioned walks in the rain, thinking about Willoughby). She decides to try and incorporate more practical elements into her life, and resolves to be more like Elinor. Her decision to marry Colonel Brandon is the ultimate practical step in her life – though she didn't immediately feel sparks with the Colonel, she intellectually appreciates that he's a good guy, and he can provide a wonderful life for her. We're not saying that Marianne just completely gives up on feelings – after all, Austen claims that she learns to love the Colonel before she agrees to marry him – but she's certainly wrapped a good, thick insulating layer of sense around her raw, emotional core by the end of the novel.