COVID-19 - Should I start applying for other jobs having accepted a post-doc verbal offer months ago?
You should only accept formal written offers.  If you follow that practice, you will never be in this situation.  

Since it is too late for that, you should request a formal written offer immediately.  Yes, you should look for alternate employment.  There are several possibilities:


You never get the formal offer.  In this case, get a job elsewhere.
You get the offer and can start the job.  Then start the job.
You get the offer and cannot start the job because you cannot get a visa.  Since you do not have a visa, even if you accept the offer the agreement is invalid.  Find a job elsewhere.
You do not get the offer, but do get a job elsewhere.  In this case, inform the university that, if you do not get the offer in a reasonable period of time, you will no longer be available.
You get a job elsewhere, and after a long time you get the written offer and visa.  Now you have the opportunity to experience both jobs!