Unless it had three eyes or more do not expect any peony to "do well" next spring. It will take at least another year before it will start looking like most of the peonies you see in other gardens. There is nothing for anyone to do with this peony other than let it do what it wants to do. Some would suggest removing the spouts but the plant will do what it wants to. If you remove the sprouts it may force the root to form new buds rather than sending out feeder roots. FYI Most peonies bloom buds are formed in a three year cycle. The buds that will bloom were formed three years ago. It takes that long from erupting from the root to becoming mature enough to form a flower bud.

Having them wake up for a bit in the fall happens occasionally; it's happened before in my garden. Typically we have another cold snap within a few weeks and the growth dies back, have never had to cut anything and can't tell the difference between those that sprouted in the fall and those that didn't as far as spring growth goes. I wouldn't worry about it if I were you.