E320 briefly overheating

The AC on my 97 E320 began blowing hot on the highway, so I glanced at the temp gauge, which was maxed. A few minutes later, the AC began blowing cold again, but the
temp gauge appeared no longer functional, it was at the bottom. I'm considering just replacing the thermostat, to see if the problem goes away, rather than leaving the car for service. Is this a reasonable course of action?
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Would consider checking the engine temperature sender - not sure where this is located- but with an erratic temp gauge the sender is the most likely cause. I would also assume the information from this sender has an influence on the AC.
Joe

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