There are a few things that may be confusing here.
1. "He was very sad when she met someone else." is normal and correct (if "met" is an event, as in "kennenlernte"). It indicates that the first thing mentioned followed (on) the second, often implying causality.
2. It would be somewhat unusual in this example, but if two actions are contemporaneous (but unrelated) you can use "when" or "while", often with one or both of the verbs in the progressive (-ing), depending on the meaning.
"She was seeing someone else while he was fighting for king and country.".
"She met a lot of men while he was working abroad".
The possibilities depend on things like whether you are talking about single events, states, repeated actions, etc.
"As" can also be used for contemporaneous -- or nearly contemporaneous -- actions (in the wider sense), but the meaning is a little different: "As he entered the room, he noticed that someone was following him.", "As he was putting on his coat, he suddenly realized . . .", "They looked up as he entered the room.".
3. "As" can also be used to express a reason of an incidental nature: "As he had nothing better to do, he decided to walk into town.".