Since this is an engineering forum, I'll say that you could add an LDO regulator to your existing adapter- using a part such as the Diodes Incorporated AP1186T5-33L-U. It should have a small heat sink to be safe (1 square inch of copper is probably enough). That will take care of the variation in your adapter output without dropping too much voltage at full current. They're $1.58 each in singles. This part is on the way out so, but for a one-off it's not a problem.
It needs a few electrolytic capacitors (the can type) in addition to the chip (3 x 100uF will work).
Personally, I'd probably just buy a switching adapter (make sure they have genuine safety-agency approval or listing markings.. not just a generic PRC "CE" mark that has no real value)-- some of the ones on eBay etc. are criminally bad. That will use less vampire power so your electric bills are less (assuming 24/7 operation this can be a factor- the type of adapter you show typically runs noticeably warm and you're paying for that wasted heat).