Congrats on the new pool! A couple of initial notes: you aren’t supposed to add salt to a new pool for the first 30 days, so you shouldn’t be using the SWG anyways for now. Hopefully you haven’t already added the salt.
Your calcium hardness is low.
While chlorine use isn’t high in winter, your chlorine level is almost nonexistent. Once algae gets a foothold in your pool, it’s A LOT of work to get rid of it, so you’re smart to address this now.
Now onto your issue: I suspect your tablet isn’t dissolving very fast because the water is cold, and there isn’t any water moving through the floater. Add more tablets and find a way to keep it in front of a return jet for water movement.
The better, easier way to manage chlorine (until you can use your SWG) is to use liquid chlorine. You can find this at Home Depot or Walmart. One quart of 10% liquid chlorine will increase your chlorine level by around 2 ppm. I would go this route because it has an immediate effect, unlike the slow nature of tablets.
As a heads up, when the weather gets warm, you’ll want to increase your CYA level to around 70 to help protect your chlorine. Your 30 is fine for now.
And finally, you should get your own test kit rather than rely on a pool store. Their readings are notoriously inaccurate, on top of giving terrible advice.