If they didn't pull up the carpet, they didn't really check. They only way to check to see if there is mold/mildew in the carpet is by ripping up the old carpet and padding. Keeping in mind that after a sewage backup, that should have been done, anyways...especially if it affected the whole carpet. There are a lot of other health hazards caused by sewage backups that have nothing to do with mold.
Now, that being said...make your final request to your landlord, in writing, about the problems in your apartment. Don't "threaten" to do anything...just put the situation in writing, and ask for a satisfactory resolution. Make sure you tell the landlord that you feel that your health is being affected by this situation. Send that letter to your landlord via Delivery Confirmation with Signature Confirmation on it--make sure you keep a copy of it. I believe he/she has 14-30 days remedy the problems (local laws vary). If he/she does nothing, then you contact the local Health Department. They will send someone out to assess your unit and write up a formal complaint. If there is something that needs to be done, they will put it in a demand to the landlord. They will give the landlord x amount of days to complete what they tell him/her to, after that they are taken to court and assessed fines for every day they do not do the job.
It is important that you not let this "slip" by. It is important that once you start this ball rolling, you see it through. That means calling the Health Department if these things are not completed. That means keeping $$ totals of all money you've spent in regards to the health issues of the condition of your apartment.
At this point in the game, you really need to get a mold remediation specialist...not just a "good" carpet cleaner. A carpet cleaner cleans carpets. But, a mold remediation specialist can tell you if/where the mold is...how bad it is...and tell you what you can do to remedy it (well, your landlord). You may have to get all new furniture, new carpet, new everything...you'll most definitely need to have the air intake/output ducts in your unit professionally cleaned, as well.
Good luck!!