First of all, get the name of the person you are talking to at the mortgage company. If they are telling you that they will transfer the loan into your name after they’ve begun the foreclosure process, as long as you make payments on the mortgage, that would be unusual and I would be leery. Foreclosure would take your name off the deed. And since you’re not now named on the loan, they would have to underwrite a new loan in your name and put you through some kind of an application process. So, if they are giving you advice about what to do and you are relying on that advice, they may be opening up a liability for themselves that you could use as a defense against foreclosure. Get the names and locations of everyone you talk to and document everything (including dates).
It sounds like, from what you’ve said above, that you are dealing with the loan servicer, not the actual lender. If that is the case, you may want to find out who the lender is, and contact them. Again, document every conversation in detail. Let the lender know that you have paid off all the back taxes and you can continue to make monthly payments. Ask them about re-aging the loan, or making a payment plus each month to get the past due amount caught up.
Depending on the state you live in, there may be foreclosure prevention funding available. For example, in Oregon, because it is one of the “Hardest Hit” states in the recession and housing crisis, the Federal government has allocated grants to help people in your situation. The current application of those grants in Oregon is to make mortgage payments for people for up to a year and catch up past due amounts. You would have to apply for the funds and meet the minimum qualifications. Contact your state housing administration or state housing finance agency and ask about the “Hardest Hit Fund”.
It is always a good idea to consult an attorney in these situations. That would be my strong recommendation. I know it can be expensive and you may not be able to afford one. Contact your state and local governments (city, county) and ask about legal aid. There may be assistance available in that regard, too.