You can continue to apply until you would be over the age 23 on July 1 on the year you would enter USNA. There are some who have entered USNA with a degree from another college.
However, IMO, at some point you have to weigh the cost/benefit of earning a commission through another route (e.g., NROTC, OCS) vs. "starting over" at USNA. Also, if you continue to be turned down by USNA, you need to consider what you can do do "change" your package. Continuing to reapply with essentially the same "qualifications" is unlikely to result in a different outcome.
As a general rule (as noted, there are exceptions), it's unusual for someone with more than two full years of college to start at USNA. However, you are certainly able to keep trying if it's your goal.
NROTC is definitely not easier than ROTC and could be harder. Part of it depends on the college to which you are applying, the number of NROTC slots there, and thus the competition. Just like it can be easier to obtain a nom (and thus an appointment) to USNA if you live in North Dakota vs. Maryland. It's NOT that certain folks are less qualified than their peers only there are fewer of them per geographic area or college and thus the sheer numbers game makes it "easier" -- although both USNA and NROTC are never "easy."