This is an excellent question, as you are very much correct that the viruses responsible for cold sores can be very, very contagious. In general, if your girlfriend was only exposed to your chapstick after the cold sore scabbed over, then it is less likely that you were actively shedding virus at the time and thus she is much less likely to pick up this infection. However, it is certainly still possible and she should discuss this with a primary care physician
. In general, it is best not to share chapsticks for exactly these reasons.
If you or your girlfriend have any questions, your primary care physician will be able to discuss this in more detail. In particular, it may be important to know whether what you are describing as a cold sore is actually an outbreak of the herpes virus HSV-1 or HSV-2. These highly contagious viruses can also cause significant discomfort if they infect skin in the genital area. For people who develop multiple recurrent episodes of either oral or genital HSV, suppressive antiviral medications such as acyclovir may be effective in preventing the frequency and severity of outbreaks. Your physician can help you determine whether this might be an issue for you. Hopefully the cold sore you have will be completely gone in no time!