Question about op amp DC offset
I'm assuming you're using a non-inverting configuration.

First, your DC offset. When you looked at the output, you had no input at all - not even a resistor to ground. So the input floated and the output went to the rail. Try connecting your input to ground with, let's say, a 10k resistor. If, for some reason, you are using an inverting configuration, ground the input. If you stll seem to have a large offset, particularly one which seems unsteady, look at it with the scope and make sure the output is not oscillating.

The reason the offset is dependent on the value is that the op amp is trying its very best to produce an output at V+. If you raise V+, the output follows it.

Next, oscillation. Take a look at the data sheet http://www.ti.com/lit/ds/sbos070b/sbos070b.pdf, Figure 1. Are you doing ALL of this? And do you have a good ground plane to connect those capacitors to? Those capacitors are there for a reason. Plus, you need a solid ground, with the power supply ground connected to the motor return right on the board close to the op amp. If you run separate leads back to the power supply you may well be forming a ground loop.

Your scope trace is pretty classic for this sort of op amp. The positive output transistors, being NPN, are faster than the negative PNPs, so a setup which is marginally stable for negative swings is unstable for positive swings.