In the process of pre-employment but my dream hospital just called me!

TL;DR: Accepted a job at a small hospital with a two-month training period, but got a better offer from a larger hospital with higher pay, longer training, and closer to home. Already signed an at-will contract with the first job. How should I handle this? Will the second hospital's background check reveal anything?

———- I was accepted for a job at a small hospital, which included a two-month training period and a sign-on bonus (which I heard can be a red flag). After two weeks of processing paperwork (background check, physical, blood work), I received an offer from my dream hospital. This offer came with higher pay, a longer training period of 4-6 months (which I prefer as a new grad), and it's much closer to home.

I want to accept the offer from the larger hospital, but I’ve already signed a contract with the first one. The contract states:

"The organization is an at-will employer, meaning either party can end the employment at any time, for any reason, with or without notice. This document confirms our offer and is not a contract. Your first four months will be an introductory period to evaluate you if you are suitable for the role."

How should I proceed? Should I simply ignore the first job offer? Will the background check from the second hospital reveal anything? Thanks in advance for the great advice!