I believe they are the same word, although with different patterns of usage, as explained above by @Anonymous123.
If you examine the kanji, you can see that 混 is more about the discombobulating confusion of a crowd, and 込 is more about entering into an enclosed space, so you might want to choose which character you use with that in mind.
That said, I’m a little confused about how 込 is a kokuji. Why was there a need to invent a new character?
Kokuji is the name for a character 字 invented in Japan 国. Most characters come from Chinese, and have both Japanese and Chinese readings (kun’yomi and on’yomi). Kokuji only have kun’yomi.
As far as the sentence goes, I don’t really know. Probably either one is fine.
