Does it matter if I don't use all five of the remaining buffers on the 4049?
You don't have to. Generally, it is good to terminate them but that's not a must.
I've attached a schematic of my shift register pcb's -
That's a layout. Post the schematic instead.
Also, the circuit for the 4049 has no resistors or caps on it - do there need to be any?
It is good to decouple. But not a must.
Your issue is likely somewhere else. But without posting a schematic, it is hard for others to help you.