Should I get a balance transfer card?

Hi all. Finance novice looking for some help, any would be appreciated. I posted in r/personalfinance and was hoping to get some additional perspective:

-34M

-Married

-Home mortgage since July 2024

Had 1st child a month ago and want to open a college savings fund

-My AmEx is closing in on 15k balance. Most of the balance came from moving things around to help make the down payment for our home. Other miscellaneous and unexpected expenses over the past year didn't help

-I have 100k in student loans which I admittedly haven't actively tackled since the gov't put them into long-term forbearance

-Gross annual salary is 92k

I pay about $1000 a month on the CC balance but I've come to realize a lot of that goes into the ether due to the high APR. Yesterday I got bothered enough by the debt that I did some research for quite literally the first time. Again, novice here (and perhaps a bit irresponsible with finances). The concept of a balance transfer card seems pretty straightforward, and a 21 month 0% APR card would be the easiest option for paying it all off comfortably, I think. Credit is high 700s so I hope I'd get approved.

The folks in r/personalfinance informed me that not only would I likely have to pay a transfer fee, I might not be able to transfer the entire 15k. I still think it would be in my best interest to transfer as much as possible and to stop touching my AmEx unless I absolutely have to.

It was suggested to be that it's unwise to open a college savings fund until I pay off the CC debt. I don't see why I should hold off doing that if college funds don't need to be added to at a fixed rate.

Finally, I am planning to improve my budgeting so that I'm not, as some informed me, living so much above my means.

Is this a good route to take? Any suggestions or anything I'm missing?