I married a B”T and have stated the challenges that we faced, in previous posts. However, I am the wife, from a FFB home, and there IS a difference whether it is the husband or the wife who is the B”T. Since the mother is the primary caretaker and teacher of the children when they are young, the boy’s family wants to be sure that she will make the proper Jewish home for their son and aineklach. That is a fair concern, and I mean no disrespect to you by saying that. I have seen the damage that can be done when the person in charge of the kashrus of the home (both food and Taharas Hamishpacha), is either ignorant or ill-informed in the proper halachos. That is another fair concern.
Sometimes when a girl becomes frum, it is harder for her than for her male counterpart when her family is not supportive of her decision to be frum. It is easier for a boy to move out, than for a girl. People tend to look at a frum girl who live on her own, with a little more suspicion than they do of boys who do the same. THAT, is not fair, but it is what it is.
Lastly, if a girl is a B”T, it is possible she may have lived her life in a way that causes problematic issues with certain types of shidduchim, like Kohanim. Another example, is it 100% certain she is halachically Jewish (is it certain her MOM is a Jew)? With the intermarriage rate what it is today, and people who are not frum, really not knowing what constitutes being an halachic Jew, this could be an issue.
Was she promiscuous in her former lifestyle? (I am not chalilah suggesting you were, but you asked what some people are concerned about, and potential STDs are a biggie). People are also worried about B”T flipping in and then flipping out. I personally know a few people who fit that description, and it is very sad.
Your best bet, IMHO, is to meet other B”T (maybe through Rebbetzin Jungreis’ Hineni group, or through NCSY, or another Kiruv organization). At least you can both grow together in Torah, and you would have the distinct advantage of understanding the other person’s mindset. I hope you find your zivug very soon, and wish you much hatzlacha.