So I'm writing a story during the course of which my main, first-person-POV character learns another language, one used heavily in her surrounding. This generally works, but I regularly get stuck on dialogue for other characters when my narrator can understand some but not all of what they're saying. I have two main approaches I use, but both seem to have real disadvantages. Any ideas?
Approach 1: Paraphrasing
A stream of words followed. My best guess was that she was talking about how much she liked the food, but given that I could only understand one word in four I couldn't dismiss the possibility that she was explaining how her day had been, complaining about the noisy neighbours, or proposing marriage.
I sighed and intoned the words that had been among the first things I learned in class. "Slower, please?"
"And I thought you-" The rest of the sentence was obscured by grammar (this language made talking about things one didn't believe true unnecessarily complicated, in my opinion) but I thought I got the gist. He'd thought I was an honest person, and now he was deeply disappointed in me.
This is great for getting across how the character is actually experiencing things, but the fact that there's almost zero direct speech has (in my opinion) a distancing effect that makes this hard to manage for more than a few paragraphs here and there.
Approach 2: Italicised foreign words
"She can't make it, she told me earlier today. She has Verpflichtungen-"
"-obligations," Sarah translated,
"-in town which will beschäftigen her until late. She wants to meet up another time," Andrea added while I was trying to disentangle the unfamiliar word.
This is a lot easier to keep up for a whole conversation, but I worry that the readers will get annoyed at the onslaught of italicised foreign words. It's also not really a great depiction of what she's experiencing, because it's hard to show things like her only grasping bits and pieces of what's going on, guessing to fill in blanks, or trying to untangle unfamiliar grammar - in this particular example, I think my character is coming off as more fluent than she actually is.
(Note that although I used German for the sake of the example, the actual story uses a conlang so at least there will be no readers who know the language to worry about!)
Does anyone know of any other good ways of showing native speaker dialogue through a POV character with an imperfect grasp of the language? Or am I stuck combining these two approaches and trying to avoid scenes with lengthy foreign-language conversations until she's a bit more fluent?
EDIT: To clarify:
My problem is not that I want to include foreign words but still have the readers understand the dialogue. In fact, I'd happily do away with the foreign words entirely.
My problem is that I want to show a piecemeal, patchworky understanding of what is said - putting a percentage on this is difficult but let's call it 40-60% with outliers in either direction. I can manage this well for short scenes via the paraphrase demonstrated above, but I'm not happy with its effect in longer conversations.