How to fix memory corruption on 2011 MacBook Air?
You have several options to fix this issue:


Have the board professionally repaired.  This is a common repair that can be done, but it's not economically feasible because the cost of the labor is very high in comparison to what you can get a new MacBook Air for
Swap out the logic board for a known working one (available on eBay for less than $200USD).  This may be a viable option, but keep in mind, your MBA is 8+ years old.  You will be expending money to get another 8 year old machine.  
Buy a new machine and transfer your data using an external enclosure.  The 2011 MBAs had a removable SSD that you could remove and put into a replacement machine or you could put it into a USB enclosure so you can extract your data.  


Personally, I would buy a new MacBook, MacBook Air/Pro.  I would remove the SSD from the dead machine, put it into an enclosure to extract the data and then sell the 2011 MBA as salvage parts.


  If I buy a new device I would still have to pay for data recovery and
  transfer service, as the old device is non functional


This is what Time Machine backups are for and it's extremely cheap insurance in case of a failure (see the section titled  "A Word About Backing up Your Data").  That enclosure could then serve as a USB drive for future Time Machine backups.