The schematic wouldn't be much help to you as I was driving a synchronous motor.
Your design wasn't that far off the mark.
You had a frequency generator, the 555. Unfortunately it wouldn't have been accurate enough.
There is a device to create a sequence to drive the stepper motor, the CD4017. It just needs adapting to the correct sequence.
There are the drivers for the motor phases. They look fine.
You could add some OR gates, or use diodes to give you the same effect, and still use the CD4017. If you "OR" the 0 and the 1 for to drive L1, the 1 and the 2 to drive L3, etc, that would work.
For the better scheme, move the reset to the 8 line, then wire it so that 0, 1 or 2 will drive L1,
2, 3 or 4 will drive L3
4, 5 or 6 will drive L2
6, 7 or 0 will drive L4
If you pick the crystal frequency correctly, you could use a CD4060 or two to divide the crystal frequency to give you the speed you want.
It is difficult to have a finely adjustable and very accurate frequency. You either need a good phase lock loop or direct digital synthesis.