I was laid off last summer and recently found a job. During my unemployment, I fell behind on my mortgage payments. My mortgage company now wants the past-due amount in full before they will consider a loan modification. I don’t have that kind of money, but I can make my regular payments. If I continue to make my regular payments, will this help me avoid foreclosure?

Only if there is some profit in doing so to them. Unfortunately, most are a scam. They will get an upfront out of pocket fee from you & then you will find yourself in no better a position than you were before you paid them their scam fee. (How they make money & stay in business)
You might be better off to call your lender and ask for a loan modification. If you don’t qualify for one now, ask what it will take to qualify for one.

Please tell me any experiences you have had with this or any other good programs that are similar.

I am not sure where to start on trying to get a loan modification so my home does not go into foreclosure. Are there any free programs or agencies that can help me get started?