Shootthrough (or cross conduction) is a problem in any bridge (whether BJT or FET). It's when you turn off and turn on a pair of transistors in a half bridge at the same time, and because transistors take longer to turn off than turn on, you get a point when both transistors are on that produces a short-circuit.
You can control this by independently controlling all the transistor gates so you can shut one off, wait a bit, then turn the other one on. THe time you wait to make sure the first transistor is off before turning on the second is called dead-time. Your circuit uses a single signal to control both of them so you can't do this. But I think if you place the right gate resistors, you can slow down the time they turn on at (and turn off as well) to make it less of a problem. I don't know much about sizing the resistors or exactly where to put them, but someone else probably knows more.
You might want to place diodes across each MOSFET too (I'm sure you know why by now, please ask if you don't). But at 5A and such a slow switching frequency you may be able to get away with using the parasitic diode inside your MOSFETs (as shown in your schematic).
WHy does your motor have 3 pins and one goes to ground?
Other than that, I don't see any problems with your circuit (except that your resistors might be need to be sized a bit differently). Give yourself more credit!
EDIT: I think you have a bad connection on the bottom right transistor's pin 2 where the wires jump over each other. I think it connected on it's own when it shouldn't have.