Once you cut down the trees, you should have the main stumps dug up or ground out. It is possible that new, small trees will emerge from the roots left in the ground. All you can do is repeatedly cut them off at the ground, or apply a nonselective herbicide (glysophate) to the young plants when they are actively growing. Be aware that nonselective herbicides will kill any plant they contact, so protect any valuable plants in the area from spray drift.
There is no way to avoid some disturbance to you gardens when removing such large plants.
