I assume you have tried to use the Powerful Cleaning cycle. If not, you should give that a try before moving on to other remedies.
There are numerous pages on line about cleaning clogged heads. They tend to use Windex (original formulation) or Simple Green, which can dissolve dried ink. There are also commercial outfits that sell kits to clean the tubing and heads by forcing a liquid through the system from the cassette end. These usually involve taking off a lot of panels to get at the ink tubing.
I was frustrated with the Epson Owners Manual because it didn't say how to free up the print head, which is needed to clean it. So I bought the Service Manual from 2manuals.com for $12. The manual is quite thorough but does not allow one to search the text, unfortunately. You might consider doing the same, since you are trying to avoid spending over $1000 for a repairman's visit.
For me the most important piece of information so far is how to unlatch the print head. Here is a brief synopsis, in case you want to try it. It is the first step in the process of cleaning the print head with Windex, which applied to several layers of paper towel.
You first need to turn on the printer in the Serviceman Mode. This is done by turning the printer on while simultaneously pressing down the OK, Down, and Right Buttons. Easy.
Next, to unlock the Carriage Unit (CR): Select Self Testing -> Mecha Adjust -> CR UnCap; Press OK.
The print head should now be free to slide on the rails so you can pull it out from its garaged position. If this procedure does not work, you may have to release the CR manually, which means taking off the Right Top and Right Side Panels to reach a latch located behind and below the CR.
I hope this helps. The Service Manual goes through all the above steps in detail and is worth the price if you plan to do any work on the printer yourself.
I am located 100 miles from a service center and so am motivated to try to fix things myself. Hence the need for the Service Manual.
Incidentally, I have had problems with my jets clogging, which I think is a result of my living in a dry part of the country. I have been placing a high-walled pan half-filled with water in the printer cavity when it's not in use. I cover the printer up with plastic bags. That elevates the humidity from 20% to 30%. I am not sure whether that reduces the clogging, but I hope it helps. I usually find that the basic manual cleaning of the specific jet, applied once or twice, suffices.