Hi Dear,
Welcome to HCM.
Understanding your concern.
Thanks for your query . Sometimes discoloration and 
redness occur after waxing because of two reasons .
- first wax is hot enough to cause redness .
- plucking hair too hardly.
As skin around lips is sensitive , it is pron to 
injury . The darkness is caused by post inflammation hyper pigmentation . 
I would suggest you to apply kojivit cream twice a day. It contain 
kojic acid , which effectively treat hyper pigmentation and apply sunscreen with SPF 15 to prevent further pigmentation due to UV rays of sun  . 
If condition persist after 2-3 weeks then 
consult dermatologist for proper evaluation of the condition anf treatment .
Doctor may recommend chemical peeling or microdermabrassion depending upon the severity of condition.
Hope your concern has been resolved.
Get Well Soon.
Best Wishes,
Dr. Harry Maheshwari