Try a quick-drying ink from Noodler's, such as Bernake Blue or Bernake Black. I'm a big fan of the Platinum Preppy with an o-ring to seal the whole body allowing you to fill it with ink—it's dirt cheap and writes very well. I recently picked up a Pilot Metropolitan, slightly more money but again, impressive value and it comes with a converter (which is like a cartridge that lets you refill with your own ink).

You can, but you have to write in that weird lefty trying to be a righty technique like President Obama does.

http://centerforplainlanguage.org/wp-content/uploads/2010/10...

To add to what the others have said, besides writing overhand (so your hand doesn't drag across what you just wrote), look for smaller nibs, drier inks, as well as more absorbent paper. Anything that will make the ink set faster instead of staying on top of the paper.

I'm not sure what the exact problem is with writing left-handed.

I have been switching between writing right-handed arabic and english all my life with every kind of pen available, it has never been a problem. I never even noticed there's a difference.