I have a shape created with the rounded rectangle tool and when I add a stroke to it, the inside corners of the stroke are not rounded. Is there a way to make them rounded without creating 2 separate objects?
The inside corners are rounded, they are merely proportionally rounded based upon the weight of the stroke. The thicker the stroke, the less rounded the inside corners become.
If you absolutely want inner rounding to match the outer rounding there are a couple options.....
Move your stroke under the [Fill] item in the Appearance Panel.
After doing so you may need to increase the weight of the stroke, since half of it will be hidden by the fill.
Or change the stroke alignment to the outside on the Stroke Panel.