How can I handle the existing thread from a broken seam?
If the ends of the old thread are long enough and strong enough, finish them off as you would the end of your hand stitching.

If the ends are strong enough but very short you can use tricks to tie the ends and tucked out of the way. My most used method is to stick the needle into the fabric close to where the last stitch is.
Then I make a loop around the needle and insert into the eye of the needle, in such a way that the thread will make a knot when the needle is pulled through the fabric. If you have enough length, repeat once or twice, to make a more secure finish.
If it is the last tuck, have the needle go up and down a few times going through the fabric of or near the seam.
You can do this instead of tying onto the new yarn, but your method will also work if the yarn is still strong enough.

I would still stitch the new seam over the old one, making the switchover more secure.

If the yarn is likely to break you may want to replace a bigger part of the stitching, even when the thread is not breaking yet. You can stitch over the old stitching or just next to it, or you can take out the old stitching and replace it with the new seam.