If you do decide to pay the mortgage off, the safest way to do so is to put the extra payments aside (invested in CDs, bonds, and/or stocks, depending on your timeline and risk tolerance) and then pay off the mortgage when you have enough saved to pay off the whole thing at once.
As others have pointed out, paying extra to the mortgage puts you at risk of losing the house and some or all of your equity if you lose your source of income. If instead, you have a mortgage payoff fund that you can use to keep making the mortgage payments, you can prevent being forced to sell the house at a loss.
The only reason to pay extra to the mortgage is if, psychologically, it's the only way you will save. And in that case, you need to have a very large emergency fund (>6 months) to protect your investment in the house.