Brenda, your peaches that died back to the graft and came back thorny were likely grafted onto plum rootstock. Peaches aren't thorny, but some plums certainly are. Around here I have a number of peach
trees that grew up from discarded pits. Most of them have good to decent
fruit, but an occasional self-sown tree will produce fruit that doesn't ripen well or isn't sweet when ripe.