The 74HC175 is a quad flip-flop in a 16 pin package. Don't know about availability.

It looks like they have both this and the CD4017 at my local store. But the 74HC175 has only 1 reset pin shared by all 4 flip flops. Doesn't this just reset all four of them to low?

 

It looks like they have both this and the CD4017 at my local store. But the 74HC175 has only 1 reset pin shared by all 4 flip flops. Doesn't this just reset all four of them to low?

it does. So, if you want to do initialization, it has to be done synchronously with the D inputs.

 

You might consider coupling an the IC with the four FF with another IC containing the next-state logic. That could probably be done using a PAL (Programmable Array Logic). An example (which may or may not work for this design) is the 22V10 or similar. This would allow you to keep the implementation to two through-hold ICs (for the core logic).