Why is my amplified strain gauge output so unstable?
it ranges by about 20 mV peak to peak when the strain gauge is fixed
  in place. And it is not a stable oscillation; it will just slowly move
  around that range, changing by about 2 mV per second.


Referenced to the input that is 20mV/567 = 35uV and 3.5uV. Your bridge has 5/(357.5*2) = 7mA passing through each leg, so a voltage change of 35uV corresponds to a resistance change of 35uV/7mA = 0.005&ohm; in any bridge resistor or connections to it. 

In the comments you say:-


  My wheatstone is just resistors on a breadboard


If this is a plug-in breadboard you should know that they are notorious for having poor connections. Your bridge and amplifier circuit should be built on a proper PCB or matrix board with soldered connections, and the sensor wires (if not also soldered) should be connected via a low-resistance plug/socket or screw terminals. 

Minimum recommended power supply for the LM741 +-10V, but it will probably still work OK at +-8.2V. More concerning is that it does not have a specification for offset drift! So 3.5uV/second drift may be quite normal while the op amp is warming up. If this is unacceptable then you should use a low drift op amp such as the AD708 (offset voltage drift typically 0.1uV/&deg;C). You might also consider using a rail-to-rail op amp which can operate off a single +5V supply, such as the MCP6V01 (offset voltage drift 0.05uV/&deg;C max).