Welcome to our Forum and thanks for your questions.  In reading your question several times, I have several comments.  I hope they will be helpful.

1.  You don't mention any exposures you might have had before your diagnosis (i.e. numbers of partners, their gender, or types of sex acts), nor whether the health care provider you saw looking for white blood cells in your urethral discharge or not.  Certainly the combination of urethral discharge and burning on urination are sufficient to make a diagnosis of NGU following an exposure and testing for gonorrhea and chlamydia are the right tests to do.  In about 40% of men with NGU, tests for gonorrhea and chlamydial infection are negative.  most such infections do respond to treatment of the sort you received but there are some treatment failures (see below)

2.  You describe at least a partial response to therapy.  In some persons, following the diagnosis of urethritis, heightened vigilance looking for signs of continued infection leads them to notice the small amount of normal genital secretions that are variably present in most men.  These secretions are variable in amount, typically most noticeable first thing in the morning and tend to be clear, not cloudy.  Is this possible?

3.  In recent years, a new bacteria, Mycoplasma genitalium, has been described which may not be effectively treated by either azithromycin or doxycycline and required different antibiotics for successful treatment (moxifloxacin).  When you see your doctor, I would suggest that you request a urine test for Mycoplasma genitalium (AVOID testing for other sorts of Mycoplasmas such as Mycoplasma hominis which can normally be present in the genital tract). 

These are my initial comments. I hope you will find them helpful.  We do allow for up to two follow-up questions for clarification so if anything I've mentioned is unclear, please feel free.  EWH

Thanks for the additional information.  Comments:

1.  Urine can be cultured and can be tested for WBC.  I am not sure which test you had, or both.  If this was NGU, I would anticipate that there would be WBC detected in your urine test, particularly if it was a test performed on urine collected just as you began to void (not mid-stream which is good for detection of UTIs but not testing for WBCs).  Classical urine cultures which look for bacteria are typically negative in men with STI-related urethritis. 

2.  The timing of your exposure and the onset of symptoms certainly suggests an STI.

3.  I agree, a substantial amount of cloudy discharge is not just "hyper-awareness".

4.  since you improved somewhat with the doxycycline, it may be worth trying again but I still think I'd test for M. genitalium before I went to a urologist.  If the test (again, collected just as you begin to void) is positive, I would treat with moxifloxacin. 

Hope this helps. EWH