Yes to running the pump. We need to know the volume of your pool. You say that your reading is 5 which is high. That is the chlorine reading rather than the stabilizer reading, correct? Although most cheap testers show 5 as being high, it isn't really all that high. Your chlorine level is dependent upon your CYA level. (See the Best Guess Table link in my signature below.)
Do you have a test kit or are these numbers from the pool store testing it? if you don't have a kit, get one. We recommend the Taylor K-2006 or 2006C which you can get from the Amazon link below. Only buy if the seller is Amato Industries, however.
Your pH is ok at 7.2. Just don't want it to go any lower. Keep it between 7.2-7.8. Your alk is fine at 80.
Please don't let the pool store sell you a bunch of stuff to throw in your pool. Pool store goop usually makes the problem worse. All you need right now is chlorine. I suggest that you just use plain, unscented household bleach for your chlorine. I use generic Walmart Ultra bleach. If you tell us your volume, we can suggest how high to take the chlorine to clear the pool. I hope this is a big AG pool because dumping 5 gallons in is a lot! If it was teal and now is cloudy, then your algae is dying and you will have to let the filter remove it. I assume this is a sand filter? Watch your filter pressure and when it rises between 5-10 psi over the clean filter pressure, backwash it. Having said that, I need to ask how you added the stabilizer? If you added it to the skimmer, you don't want to backwash for at least a week after it was added to give it time to totally dissolve. We usually suggest waiting until the pool is clear before adding CYA.
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