How long does it take for ear piercings to heal?

Soreness during the healing period of a piercing is completely normal. If your piercing was infected it would be producing a darker yellow/green discharge in large amounts.

You need to stop with the aftercare you're doing. Twisting a healing piercing drags in crust and bacteria and tears up the inside of the wound. Throw out the crap they gave you - it's far too harsh for a healing piercing. The proper way to clean a healing piercing is by doing saline soaks twice a day for 5-10 minutes each soak with a sterile gauze pad soaked in a sterile saline wound wash aftercare spray which can be found at any pharmacy.

Claire's is awful and sorry to say but you should not have gone there to get pierced. Instead of 2-3 months of healing had these been done with a hollow needle, you're looking at 4-6 months. It would be a really good idea for you to visit a reputable professional piercer to at least get the crappy unsterile low quality jewelry replaced with something that is sterile and implant grade.

Leave it. Mine had all sorts of swelling and issues and I just left it, clean and twist it as appropriate, it will be sore for about 2 weeks and after 6 weeks, you can take them out. If by the 6 week mark it's still swollen or has liquid around it, see a doctor about antibiotic cream as its probably infected. Infections usually heal themselves as long as they're cared for properly.

yeah make sure not to take it out, keep cleaning and twisting it. The pain will eventually go away.