Can I apply to other postdocs after accepting an unofficial postdoc offer?
Till I get any official letter can I apply to other places also ?


Depending on the laws in your country, it is likely that you are allowed to do so without breaking any law. However you should be very careful about damaging your reputation in case you end up committing to several positions. 

In my experience, in academia an unofficial commitment can be trusted and official paperwork can take some time to be issued. This depends on many things obviously, in particular how much you trust the PI who offers the job. 

Since you mention that your main concern is about having a backup plan, I would recommend inquiring with the future PI (or the administration of the future institution) about their administrative procedure. Why not also mention to the PI that you are considering other offers? This could motivate them to speed up the process. And in case you feel that they are stalling, then you have a good reason to look elsewhere.