How to fix asymmetrical face without surgery?

Firstly, be reassured that nearly everybody has some facial asymmetry. You shall probably be overly self-critical, so, get the honest opinions of those to whom you're close. 
Secondly, it is hard to give you guidance without photos, but I'd like to put the following on your radar: 
- Whilst you are probably talking about static asymmetry (i.e. difference(s) when your face is at rest), it is also important to consider dynamic asymmetry (i.e. difference(s) when you are making facial expressions); 
- Soft-tissue fillers are most likely going to provide the solution for you.  You have stated that you cannot afford fillers, but please be aware that if you don't spend enough and/ or you try to take cheaper options, then you run the risk of not getting the outcome you want despite having spent some money, and, therefore, still dissatisfied but more frustrated; 
- Other modalities for attempting to improve asymmetries include threads (e.g. Silhouette Soft), radiofrequency approaches (e.g. Syneron - Candela eTwo or Profound), and focussed ultrasound (e.g. Ultherapy).  Not all can be used in all areas, e.g. focussed ultrasound should not be used around the lower eyelids due to the risk of causing an ectropion, in my opinion. 
All the best to you!