Your air conditioner won't run for long (or at all) on a 15-amp circuit; it will trip the circuit breaker. It might run on a 20-amp circuit for a short time, but you really need a 30-amp circuit for it to run reliably. Also, trying to use a 15 or 20 amp circuit could damage the compressor, if the air conditioner will run at all.
How are you connected to the 15-amp circuit? Are you using the RV shore power cable, a 15-amp household extension cord, or an outdoor, heavy duty 15-amp cord? And, if you are not using the 30 amp shore power cable, how long is the 15-amp cord?
If you are using a 15-amp cord and trying to run the air conditioner, there is probably a significant voltage drop in the cord that results in your converter running under brownout conditions (less than 90-100 volts), which may be causing the dim lights. If the breakers on your home breaker box don't trip, the extension cord could get hot and melt, or catch on fire, due to drawing too much current through wires that are not rated to carry the load.
With insufficient details provided, most of the above are just guesses; and I suspect that there is nothing wrong with electrical hookups in your Airstream. I'd just go ahead and use the RV shore power cable to connect to the 30-amp outlet at the RV campground. In all likelihood, everything, including your air conditioner, will work fine.
If you continue to experience dim lights, the first thing I would check is your batteries. In a 2008 trailer, the original batteries would be about four years old; and it's about time for them to fail. However, before buying new batteries, I'd have them load tested at your local O'Reilly/Checker, Autozone, Pep Boys, Walmart, or similar auto parts store. (This is a free service, because they hope to sell you a new battery.) The batteries may just need to be charged fully, which may happen as a result of driving to your destination and/or being hooked up to shore power for a couple of nights.