bearisgrey, yes you can do that, but one should probably contact their County Extension office for advice or your local Master Gardeners program.
The most difficult part is the after care of the graft. Proper care has to be maintained for one to two years in order for the graft to take correctly and to maintain a healthy plant.
The time of year, where the cuts are made, how they are attached, maintenance, all play an important roll in the success of the graft.
Hope this helps. The old trees are worth saving if possible.
Good luck.