What should I do if a new employer gives me an offer, I accept it, then when I go to hand in my two weeks to my current job, my old job gives me an promotion and pay raise to stay?

First, I want to thank this sub and the other career subs for the members that provided good advice and help me land this offer. There's a lot of sh1t and AI bots you have to filter through, but there are good people who gave helpful advice and kept me going despite a lot of rejections. Thx

Now, to the dilemma, I finally did get an offer, and it's a decent enough offer that I took it and even handed in my two week-ish notice yesterday. What I did not expect is that my management come back and say they want to keep and and are willing to give me a bigger title, and a bit of a pay raise, but I would pretty much be doing the something. Bumping from an aerospace engineer, to a discipline lead engineer. The other company, also in aerospace, offered me a senior title. Not quite the pay increase that I was hoping for, especially since they are one of the big 2.

Now, I'm unsure what to do and how to proceed. Other than the obvious "make a pros/cons list", "talk to your family", and get as much information as I can, I'm not sure how to talk to the recruiting team and my current management about the situation. I feel like anything I say at this point is going to start burning a bridge and in either role, I still have to work cross-company due to the projects I will be on.

Sorry I can't give to many details publicly. Hope someone has advice I can use. *crosses fingers*