Edit: I see Davydd already replied with good advice while I typed this but I'll post it anyway.
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Plugging a 15 amp cord into a 15 amp outlet is safe. Your A/C (or other appliances) would trip the breaker if the circuit gets overloaded.
I think of adapters as follows: (this is just how I describe them - don't know if it is correct, I might have the descriptions backward
)
15A (plug) to 30A (outlet) - allows you to plug a 30A RV cord into a 15A outlet
30A (plug) to 15A (outlet) - allows you to plug a 15A cord into a 30A RV outlet
The way I think of it would mean that you'd have a 30A (outlet) to 15A (plug). Using that adapter at the RV inlet rather than the campsite power pedestal outlet like what you've purchased is a good idea
For myself, I'd try what you are doing if you have an 11,000 BTU air conditioner. I don't think I'd go so far as to recommend it though when it comes to running the air conditioner. You're more likely to have success at home with a dedicated 15 amp circuit. Campgrounds can be iffy - amp draw goes up as volts drop.
The fridge, converter/charger and microwave (not in convection mode) etc. should all be fine. The air conditioner is still going to be a maybe. Running too many things at once obviously won't work.
This past week I used a 30A (plug) to 15A (outlet) adapter at the campsite power pedestal outlet often because some campsites only had a 30A RV type outlet and I wanted to use my 15A cord.