Just buy Walmart's generic bleach (8.25% sodium hypochlorite). It is sold in 121-oz jugs so not a gallon but about 0.95 gallons each. Go ahead and add 3 of them. That will increase your chlorine by about 8. With low CYA, it won't stay that high for long, though.

Yes, you need stabilizer (CYA) in the pool. The trichlor tabs have CYA in them so will gradually add it but it takes time. If you are going to use them for awhile it is ok but continued use over long periods causes CYA to get too high so you'll want to monitor it by testing for it every couple of weeks. (No need to run that test daily plus it will waste your reagents.) Just make sure that your trichlor tabs have NO COPPER in them as some do. You do NOT want copper in your pool. It can turn blonde hair green!

Personally, I wouldn't want a chlorinator. I'd rather do one of two things. 1) Use granular dichlor (which is fast dissolving and adds CYA and chlorine) until my CYA was 50 and then switch to bleach. Or.... 2) Add CYA upfront as a separate product and then just bleach for my chlorine. Just my 2 cents.

Is this a new pool just filled this spring?

Have you tested your CYA with the 6-Way kit. (I hope it is the drops-based 6-way.)

CONGRATS on the baby boy!!!! There is absolutely nothing as precious as a new baby. Not that I don't like girls, since I am one! , but I'm kinda partial to little boys --- since I raised three of them myself!