I learned a lot of things by socializing with other people.


She mingled with foreigners as she danced.


I'm not a native speaker, so somebody please correct me if I'm wrong, but the sense I get from these words is that 交じる feels more objective and one-sided, while 交わる feels more general and reciprocal.

The image I get from 交じる is one of a lone intruder or small amount of something being inserted into/mixed in amongst a larger group or mass. So in the proverbial "needle in a haystack", the needle would be 交じっている in amongst the hay, and in your sentence about 外国人に交じって, I feel a certain emphasis on the sense that she's the odd one out, the lone Japanese person in a sea of foreigners.

交わる, on the other hand, feels like it puts the emphasis on the "interaction" part of mixing rather than the "intrusion" part. If two things 交わる, I get the sense that they are both affecting one another and becoming in some way intertwined. In that sense, I think the word "mingle" that you've used in the translation of the second sentence would actually be more appropriate for 交わる than for 交じる.

Perhaps as an extension of these distinctions, I would usually expect to see 交じる used with the に particle (since it's a directional interaction of one thing entering amongst many) whereas 交わる would more usually take と (since it describes mutual interactions). I don't know if that's a hard-and-fast rule, but it matches your examples and most uses that I found in a cursory search.

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