Remember, your break of a paragraph is on each action and dialog is it's own action. There are two ways to do this. Either Mike sighed then spoke.
"I'm sorry," he said, "I can't..."
Or Mike sighed while he spoke.
"I'm sorry," Mike said with a sigh, "I can't..."
Either way, the reason it looks bad is that you have a paragraph describing two actions that it looks like do not occur in the same instance of time. If Mark sighed as a pause before he made a statement, then the two actions need a paragraph each. One line with two words is perfectly fine, especially since you're not writing prose, but describing actions as the happen in a sequential order.
If they happened at the same time, then the reason it looks bad is that Dialog never occurs between two non-spoken segments in the same paragraph. Remember, dialog is an action, thus the text in quote is the predicate action that the Subject ("Mike") did (spoke"). So in this case, his voice was speaking while he sighed, thus the sigh has an adverbial relation to the verb (the action) of said.
A paragraph of dialog is about the action of making a quoted statement. So the quote is the result of the action (What happened? Someone said something. Who said something? Mike! What did Mike say? He said, and I am quoting, "insert quoted dialog". How did he say it? With a sigh (alternitively: N/A if he didn't sigh while speaking). To whom did he say it? Implied Paul (Answer may change if there is a third person or he's speaking to himself), When did he say it? After Paul spoke (alternatively after Mike sighed).
In addition, there are some unwritten rules. Quotes may stand on their own or be broken but they cannot break non-quotes. If you start a paragraph with narration out of quotes, the paragraph must end with quotes. If you break a paragraph by inserting the said action between dialog, you need to end with a quote as well if. If you end with a non-quotes, you must begin with quotes. Never write a quote dialog in a matter that keeps a quote-unquote-quote-unqoute or unquote-quote-unqoute-quote pattern to emerge. If this happens, find away to break the second quote into a second paragraph (you have this occur in Paul's dialog as well.) or merge into a single center statement.
With respect to pronouns, a pronoun always refers to the last subject it is applicable to. So because your example starts with Paul as the subject of the action, the opening snip of the next paragraph "He heard Mike sigh" should not occur. It's still Mike's action and if Paul is the POV, you don't need to say he heard Mike sigh. It's implied he heard when "Mike sighed" is an action. Unless Paul is going to perform a reaction action to the sigh (he doesn't) specifically, he's not the subject of this sentance because he did not sigh.
Third Person is a format where the POV and the narrator are not the same person (as opposed to First Person, which they are the same person, or Second person, where the audience is the POV but not the narrator). So in this format, while the story is from Paul's point of view, Paul is not telling the story (that's the narrator). So Paul hearing Mike's actions are not necessarily to the story, unless Paul only heard part of Mike's actions. Paul should only sense a full action of another character if that action is from an unknown actor or if it is a partial action (With Paul's back turned to Mike, suppose Mike sighs, then pulls a gun on Paul while he isn't looking. Here, Paul hearing Mike's sigh is critical because Mike's next action is hidden from Paul's senses until the gun goes "Click" or "Bang" or Mike says "Reach for the Sky!" or some other cue that the gun was pulled. But because Mike is reacting to a fully known action and it is part of his natural reaction to Mike's dialog and he is not acting out of the relative senses of Paul, it's best to let Mike do the action, rather than Paul.
And since the sigh and the response dialog are using the same sense (hearing) no conditions are being changed out of the dialog. If this was first person, then Paul would need to say "I heard Mike sigh" because he is acting in his role of narrator, not in his role of POV (which is bound by the same scope of sensory perception as the POV).