"By the time she gets to the airport, I'm going to have been waiting for her for two hours."
I was wondering if the future perfect progressive is commonly used with "going to"?
I'm not sure though what you meant by "sounding weird and foreign". They are two different things. If you speak English with a non-English accent, it will be clear that you are foreign even if your actual English (grammar etc) is perfect.
I don't know how someone could sound "weird" by their use of the future perfect progressive!
I think non-native speakers worry far too much about what native speakers are going to think when we hear you speak. Generally, we are simply very impressed that anyone has made the effort to speak English at all - we don't pick it apart or pick up on all your errors. We listen attentively and only if we really can't understand something would we ask you to rephrase something. Note: I'm talking about when you're speaking English direct in a natural, conversational situation. On here, we will correct your grammar, spelling, punctuation etc as you're specifically on this forum to learn.
You can actually use them together.
"I'm going to be a better student," I said 23 months ago.
"Next month, I will have been going to be a better student for two years." (No one would actually say this, though it's grammatical)
"Next month, I'm going to have been going to be a better student for two years." (No normal person would even think of this, but it's grammatical.)
"In four years time I'm going to have been going to college for two years," I said almost two years ago. "Next month, I'm going to have been going to have been going to college for two years for two years." For two years, I will have been intending to have been going to college for two years in two years time. (I'm pretty sure that's right! But don't try it at home!)