You do not have to create a new world to get quartz, you just have to travel somewhere you haven't been before. There are a few options in this situation (in the order I would do them, and there are probably a couple ideas I missed before moving the base):
Create a stable age and mine/put a quarry there to get Quartz. Use linking books to teleport directly to the quarry platform.
If you are having trouble anchoring in separate ages then travel a very long way in your world until you generate new terrain and can start mining quartz. Use a portal gun to travel between this new base and the old base.
If you don't have access to a portal gun then use multiple linking books and the Nether or Twilight Forest to allow jumping from Home Base to Twilight Forest/Nether to Quartz Base, and vice versa.
Move your base.
As to the issues with keeping other ages anchored. The Quarry itself will keep the area loaded, so just put the quarry in a corner of a chunk so you can put your Tesseracts in the rest of that chunk, which it will load for you. Just make sure not to use an ender chest, as it can cause issues while Tesseracts will not. Also, if you are using a World Anchor currently for any reason I would suggest Chunk Loaders instead. I find them much more reliable, and you can control what they load better.
http://feed-the-beast.wikia.com/wiki/ChickenChunks
