"Someone in another forum pointed out the "Oklahoma" series of Zinnias.
They appear to have the bloom form I'm seeking, so I'm shopping around now for trades."
If the trading thing isn't successful, the Hazzard's Wholesale Seed Store sells seeds in "wholesale" quantities, but doesn't have a minimum order, so it can be an economical source for the home gardener who needs quite a few seeds. Hazzard's has a good selection, including separate colors of Oklahoma zinnia. I was just now browsing there, and captured a couple of URLs with Oklahoma listings, but the links look kind of weird and might not remain viable.
Oklahoma impresses me as an improved version of the old "Cut and Come Again" strain of zinnias. They are also intended to be cut flowers, so they may have stems that you like. Harris Seeds lists Oklahoma
And the Harris "professional grower description" says, "Bred specifically for the cut flower zinnia grower, Oklahoma Mix produces numerous 1 1/2 to 2" semi-double to double blooms on bushy plants. This mixture continually produces cuttable stems throughout the season, making consecutive sowings less necessary. Mixture of carmine, ivory, pink, scarlet, white, and yellow. Height: 30", 18-20" spread."
Those "cuttable" stems might be right down your alley. But they still look like Cut-and-Come-Agains to me. Incidentally, you can still find some of the heirloom Cut and Come Again strain. Hazzard's has them, 1000 seeds for $4.35 or 2000 seeds for $7.70. That's cheaper than buying them by the packet.
Talk about cheap zinnia seeds. Several years ago I bought a pound of "Burpeeana" zinnia seeds. I'll probably finish them off next year, since I plan to more than double the size of my zinnia garden.
(not associated with any product or vendor mentioned)