I would propagate herbaceous Paeonies by division in Sept or March or by sowing seed in a cold frame or sheltered part of the garden in September.
Divide without lifting by carefully excavating to a portion of root and cutting pieces away with some root and at least one growth bud. Paeonies don't respond well to be lifted and replanted. If you want to divide the whole plant into as many as possible, then lift by all means.
Seeds are double dormant and need two periods of chilling. They can take up to five years to flower.

You are unlikely to have success with your water method.

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Seeds will be interesting, but remember, the flowers you eventually get will not be all alike, and probably not much like the parent plant either. But there may be a beauty among them.
Most peony varieties tolerate taking root cuttings well enough, once they have grown to a reasonable size, say some 5-7 years, only not too big, not too frequent & not too many cuts, though. If you want several plants from one bush, better lift it altogether and divide into pieces with at least 3-5 buds per root division. Sounds like a contradiction to peonies not liking to be disturbed, but so the experts say.
Some quite old plants virtually ask for division, by getting an empty middle.

I found they divide perfectly well and don't mind being disturbed in the least. The trick is do not bury your pieces of root to deep. Peonies like to be vey shallow. You can even just lay the pieces of root on top of the soil if you wish and they'll still grow although I prefer to half bury them

Cuttings of herbaceous peonies are extremely difficult. I wouldn't bother when division and root cuttings are so easy. Seed is very slow.

Gerards Herbal (or was it Krutch?) says that paeony should only be dug up using a wild dog to pull it!!
I can confirm that they are extremely difficult to dig up!

Do paeonies self-seed? I've found a couple in odd places in my garden, but as I've only had it for 1 & 1/2 years, it's possible that the stray paeonies are the remains of a once-big clump, rather than new beginning. One of them flowered this summer (not last), a white flower - the established clump is pink.

My tree peony certainly self-seeds - I throw away dozens of seedlings each year. Double dormancy - the seeds start to grow roots after the firsy winter, but only show above ground growth after the second winter.

I've bought some bagged peony roots in a sale - a bit manky, but I selected the ones with most signs of life!

There were some large broken-off roots, so I've potted these up as root cuttings. What are my chances of success with these at this time of year? Also, should I have cut the pieces into smaller sections, they are about 4" long?

if you want to grow peony,the best way is plant peony not seed.Because the seed is very slow.I know peony very much.