You would still be able to apply for a card in your own name. Being an additional cardholder on her account has no bearing on you being able to get your own account. In fact - it's a very popular tactic. Husband refers wife to get bonus, and then cancels his card. Six months later, wife refers husband etc...

So go ahead. You can be a supplementary cardholder on her account, and also get your own account in your name.