A few things to improve on for the updated Schematic:
1) The LM556 should not have +12 inputs (eg.: RC and Reset pins) while being powered from +5v, possible burn out. Much better to keep all inputs and power at the same supply level.
2) The four switching transistors (Q5,6, T9, and unmarked) need to have current limiting resistors at their base pins. Higher current switching transistors will require lower value resistors. So for Q6 I'd recommend about 390 ohm, T9 about 500 ohm, Q5 & unmarked about 5k. Later testing may reveal possible value changes depending on current flow requirements.
3) Q6 will now turn on "only" when both Q's are low, is this what is needed? Your recent comments about this do not match up so well.
a. "If either of the 556 circuits go low, I want it to cut the power to the circuit."
Defines a NAND function.
(0 1)=1, (1 0)=1, (0 0)=1, (1 1)=0
b. "If either are high, the output should be high. Then, it will turn off the PNP."
Defines an OR function.
(0 1)=1, (1 0)=1, (0 0)=0, (1 1)=1,
c. "If it is both are low, the circuit should be on."
Defines an OR function.
As written condition "a" contradicts "c" for input condition (0 0). To settle this you will need to redefine condition "a" or "c".
4) I don't yet see which supply voltage will be powering your logic gates. One way to define this on a schematic would be to include one or more by-pass capacitors connected from power to ground then list the IC's that they are associated with.
5)A good way to keep track of your logic signals is to add signal names next to the schematic lines. This way you can verify the logic just by following the signal names. For example the signal coming from S1 pin 2 could be named "S1", then later after passing through IC10 the signal name becomes "~S1" (or any way you choose to designate an inverted signal). This practice will help debug your logic during the design phase and help again during the debug phase.
6) Rather then showing unconnected lines for the motor locations it would helpful to draw in the connections for your motors. For example you can place simple I/O connectors to show where the + and - contacts will be.
7) IC2 pin 4 and IC8 pin 4, have their outputs shorted together. You need to provide another logic gate, a resistor, or some diode switching here depending on your expected logic.
Similarly IC4 pin 4 and unmarked transistor collector, are shorted together, as above you need to add logic, a resistor, or diode here too.
IC5 pin 4 and Q5 collector, same as above.
With logic gate outputs shorted together you cannot be sure if either will over power the other, still this is not a good idea (unless they have special output modes), in some cases this might burn out one or both parts. For the transistors, they will most likely over power the logic gates, but it might also burn them out.
If one output must dominate you may be able to place a resistor in the path of the other output.
If the two outputs need to be combined then use another gate to define the logic. If only simple logic switching is needed you may be able to use one or more diodes or resistors.
Overall you don't want one output to fight with the other output if they were to switch in opposite directions.