get caught up:
Meaning: to become involved in something, often without wanting to:
They were having an argument and somehow I got caught up in it.
get stuck in OR get stuck into:
Meaning: to start doing something enthusiastically:
We showed them where the crates had to be moved to, and they got stuck in straightaway.
You really got stuck into your food (= ate your food quickly) - you must have been hungry.
Mum brought in the sandwiches and told us to get stuck in.
About 'was stuck...' - your 2nd sentence:
I believe '.....was stuck in a traffic jam' and your 1st sentence means the same.