To do a release, create a new serverless function with the new code. Initially, no traffic routes to it, all traffic goes to the old function. Change things so 1% of traffic goes to the new function, and see if it looks okay. If it does, make that 2%, then 3%, or 4%, or 10%, etc. Once it's 100%, delete the old serverless function.

The provider hosting your serverless functions might not give you that much control over routing. In that case, you would need to write another serverless function which sits in front of your serverless functions and does the routing. If you ever need to update that, well, then you have to cross your fingers.