How many welds show the oxidation you're talking about? Two or three? Four perhaps? In this case, I suggest to grind the oxidation by means of an aluminum oxyde grinding disk and reweld again. You won't spend that much manhours. Do not use a silicon carbide (Carborundum is the brand name) disk, because it will leave traces of carbon on the ground surface that will promote intergranular corrosion.
If the oxidized welds are, say, 15 or 20, I would consider acid pickling. Don't use hydrochloric acid because it corrodes 316L SS. Don't use neither nitric acid, it's difficult to handle due to the harmful fumes it releases. Use dilute sulfuric acid (say 20% acid and 80% water). It doesn't corrode 316L SS. Warning! When preparing the solution, add the acid onto the water, never the water onto the acid.
Don't be afraid, 20% sulfuric acid doesn't corrode SS, but if you're still in doubt, add a corrosion inhibitor to the solution. The corrosion inhibitor will let the acid to dissolve the oxydes but not the metal. Get in touch with a reputable chemicals supplier and ask him to recommend a good inhibitor.
Giovanni S. Crisi
Sao Paulo - Brazil