Travel agent here. Flights are hard for us to get cheaper than you as up to around 50 percent of it is tax. We sometimes can access different categories which will mean cheaper. And/or we get a small percentage off retail prices. Often the savings we can make for you are around knowledge of alternative flight routes and dates. Sounds like OP doesn't have much flexibility here so this is potentially why the travel agent can't help save money in this instance.
If booking yourself please only ever book via the airline itself. Not Kayak and other similar supermarkets. If somewhere is saying that the are cheaper than the airline itself then be very wary (and make sure you are comparing like for like and have gone all the way through to checkout as a PP said). Flight supermarkets often are not giving you live prices and so will ask for more money a few days/weeks after you have booked.
Flight prices very much depend on when you book them. Once the cheapest category sells out the next cheapest category is on sale and so on until all sold.
Some travel agents will add commission on the flights so will always be more expensive than buying direct. But as you say they then give you financial protection if you book as a package plus can help you out with lost luggage, sourcing and rebooking delayed flights and so on.
After transfers I'd say that flights give travel agents the most hassle.
Some travel agents have no say in what they charge, ie they have to add X percentage onto everything no matter what. But some independent travel agents have full flexibility (not all though).
I do have flexibility and don't charge anything for flights but do charge commission on hotels and cruises as I can get them much cheaper than the public so still end up saving the client money and making me money.
OP - if you do end up sorting yourself then make sure that you leave at least 48 hours between arriving at your destination and your cruise departing. That gives you the best chance to get your cruise in case flights are delayed.
Hope that helps. Happy to answer any other travel agent related questions if that's of help to anyone.