If you exit SA with an expired visa, you will be served with an undesirable notice which will ban you from entering SA for up to 5 years. You will then have to submit an appeal in your home country, which will take many months.
Rather than going through this process, you should apply to the immigration section of DHA with a letter of good cause. Essentially this involves writing an affidavit detailing the reasons why you overstayed your visa and why the overstay was outside of your control. You should also submit details of communications between yourself and the various official departments which support your claim. In addition, you must provide DHA with the rest of the documents you would provide in relation to applying to renew your particular visa.
If the DHA officer accepts your claim, they will issue you with a receipt and inform you that your application will be processed. If your application is successful, you will be issued with a Form-20 which will allow you to submit your visa application.
I suggest you do needful for another visa immediately or leave the country. Else it will be more difficult for you to obtain a visa in the future as you are currently considered illegal in the country without a visa.