I know the feeling except I did pass out when I found out I had HPV (LOL).
Your immune system is fighting the virus and a weakend immune system will make you prone to colds.
Also many people's systems can fight the virus off and with warts that can take some time. Most websites say within 2 years.
The best immune boosters I have found were changing my diet, getting enough sleep, exercise and you won't believe this but intaking yogurt. Yeah I know it sound weird but it helped keep my immune system boosted. I also doa multivitamin daily. But have confidence in your immune system, it's in attack mode and that could be why you are feeling slightly ill.
Also reduce your stress about the warts. HPV is very common and there are many many treatment options. My boyfriend has used Aldara cream which is an immune booster to help your body rid the warts and he hasn't seen a wart in about a year. Aldara however didn't work that well for me but the TCA acid worked wonders. I guess is just how your body reacts.
You may want to contact your doctor about the colds because there could be something else going on like an allergy attack with this being spring and all and allergies can feel like a cold.
Hope this helps