He must be running Carol

This phrase is said as part of the script of a film where a CIA agent who is romantically involved with this woman Carol is traveling with her and apparently going on some sort of official mission where she can involuntarily help him with that mission.

So, that's what's his colleagues guess that he is doing: "He must be running Carol".

What synonym verb would you use instead of "run" in this case?

What's exactly the idea expressed here? Does it mean that the guy is taking advantage of Carol in some way?

Thanks!

In the context of the question, run is used in the following sense:

6 d : to employ or supervise in espionage
// run an agent

In other words, synonymous verbs include employ and supervise. You could also use manage.

From "Lessons From My Father: How To Run A Secret Agent," by John Hadden:

Son: So when you’re running an agent, you feel quite responsible.

Father: Oh, God, yes, it’s terrible, yes, it’s a strain. It’s stressful in two ways. One, it’s stressful professionally because you screwed it up, and the other stress is human. You’ve gotten quite close to these people, and you…

Son: How so?

Father: Oh, Christ, you talk to them hour after hour after hour, to make sure they don’t make a mistake, walk them through each step of the way, and make sure that they feel useful, and valuable, and valued, and that what they’re doing is worthwhile, even though you don’t think so yourself … It’s hard …