Is it damages to cause someone's premiums to increase?
Yes. That increase of premiums reflects that the insurer now considers the insured less reliable (i.e., riskier exposure). Such lowering of the insured's reputation as a result of other's fault is actionable, at least on equitable grounds.
Can I sue her for frauding the insurance as well as (apparently deliberately or carelessly) causing my premium to increase?
You could sue only for the latter.
It is unclear what you mean by "[de]frauding the insurance", but only the insurer would have standing to sue her for any claims sounding in insurance fraud. At first glance, her act of signing the title over to herself (thereby excluding you) ordinarily seems irrelevant from an insurance standpoint.
You would need to check whether the insurance policy imposes on the insured the condition that the name of the insured and the name of the policyholder be the same. If that is the case, you as policyholder would be responsible for the discrepancy, since you are the one who made to the insurer any and all representations at the time of screening your profile and signing the policy.
In the event of insurance fraud caused by either you or her, it is likelier that the insurer would sue you, in which case you would need to either file what is known as third-party complaint or file a motion to add her as co-defendant in the insurer's complaint. Identifying which alternative is better will depend on the particulars of the case.