Got Shell 93 Octane and now barely idles 1973 Dart 340

Is there anything that I need to be concerned with swapping the 340 for the 360 that will not be compatible?

Make sure you find out if the correct flexplate/TC combo, and the correct crank damper, comes with the new 360, or get the right 360 parts. 360's are known as 'external balanced' (which is actually 'partially externally balanced' to be correct). This means the damper on the front end, and the flexplate and/or torque converter on the back end, are intentionally imbalanced to aid in the overall engine balance. You can't use 340 parts because:
1. The 72 and earlier 340's have no imbalanced external parts.
2. The 73 340's do have imbalanced external parts, but they are a different level of imbalance vs 360's. So if you have an imbalanced 340 torque converter, that imbalance is not right for the 360.

For the 360 front end of the engine, it has to have a 360 crank damper. On the 360 back end, it can either have a neutrally balanced flexplate + an imbalanced torque converter, OR an imbalanced flexpate + neutrally balanced torque converter. The B&M 10236 is an example of an imbalanced 360 flexplate for the 727 trans.

I hope that all makes sense....