Can the Tensioner Spring go Bad on a Rear Derailleur?
The tensioner spring is designed to keep the entire chain under tension, even after it passes the crank.  So, the symptom of a too-weak tensioner is slack in the chain between the crank and tensioner.  

It sounds like the distance between chain links is not quite the same as the distance between valleys in your sprockets.  This is not unusual if you replaced a very worn chain (because the sprockets would have worn to match the old chain).  It's also not impossible that the sprockets are worn not quite enough to match the old, worn out chain, yet still worn too much to match the brand new chain.  You may need to replace the rest of the sprockets in the cassette.