Your first paragraph is fine and best. The rule is to start a new paragraph if a new person speaks.1 In your case, you are just relaying circumstances of Elisabeth's reactions in relation to William's verbalized statements, but she is not herself speaking. The final source I give below notes the big, basic rules of needing a new paragraph (I've added numbers here for reference, they are bullet points in the source):
- When you start in on a new topic
- When you skip to a new time
- When you skip to a new place
- When a new person begins to speak
- When you want to produce a dramatic effect
Going through that checklist in your example, you are still on the topic of the family business (#1), have not changed time (#2) or place (#3), do not change speakers (#4), and then #5 is really a stylistic determination. Do you want to emphasize the groan. If not, what you have as a single paragraph is fine. If so, then possibly making a new paragraph would matter. In such a case, you would perhaps say even less to make the effect more dramatic, so:
William said, “Sweetie, raising greyhounds is not easy. But it is the family business.”
Elisabeth groaned.
William continued, “Of course, you don’t need to take over the family business.”
1 Some sources that note this: