“Nagging doubt” is correct here. However, you would not say, “the constructors.” You can say, “the construction crew has…” if you mean the team working on a construction project. You could say, “contractors” which would be commonly understood (in this case) as people working in construction. “The contractors have a nagging doubt…”

Great! Since it’s multiple people, you can say, “…investors had nagging doubts about investing in it.”

This is a good example. Usually you would say “I had nagging doubts” + something bad that might have happened. So you could say, “I had nagging doubts that my toys were missing” or “I had nagging doubts that something bad had happened to my toys.”

We use the preposition “about” with “nagging doubt.” So you would want to say, “I have a nagging doubt about one of my co-workers.” I definitely think it’s hard to be productive in an environment if you can’t trust the people around you!

Perfect! It might be a slightly better word to say, “Should I exchange my car or not?” or “Should I trade my car in or not?” When talking about cars, it’s common to “trade a car in for another.” That means you sell the car to a dealer and buy another from the same dealer.