How to get out of an offer after acceptance?

After a very long job search, I recently was presented with an offer. I accepted immediately in an act of desperation. Today, I was presented with another offer, which I feel is the perfect position with the perfect construction employer. This situation is common and the issue becomes one of character. Yes you should look out for your family's best interest, but at what cost to your character? There is no good way to back out of your commitment, even though most construction employers understand they may lose a new hire to a counteroffer or some other situation prior to a start.