The range receptacle is NOT "over capacity". It is just right.

The easiest way to check the appliances is to turn off breakers until one or both of them goes out.

It is most easy to simply move one appliance to a different existing circuit.
Next would be adding a new circuit and receptacle for one of them.

You CANNOT increase the capacity of the circuit. Circuits are wired and breakers are sized according to code. The wire used is the limiting factor. You cannot simply make the circuit bigger.

There is not another spot in the kitchen with space to move either appliance and I don't want to string an extension cord over to reach an outlet in another part of the kitchen.

If they're sharing a circuit, unless the entire kitchen is on a single circuit, I assume either the microwave or the toaster oven outlet could be moved at the breaker box to share with a different outlet (such as the outlet with the can opener or one of the outlets across the kitchen that isn't being used at all). Is this true and is this a simple and quick job for an electrician?

"I assume either the microwave or the toaster oven outlet could be moved at the breaker box to share with a different outlet "

Very unlikely.

Receptacles are not normally wired as individual 'home runs' back to the panel.

A line from the panel is used to fed multiple devices one after another.

Depending on how the house is built and finished it could be anything from a few hours of work to a major project to run another line from the panel.

What brick said.

If a separate line was run to the panel then it would already be on a different circuit.

You simply need another circuit run.