Twitter

I am searching for an app that will allow me to have both of my twitter accounts open at the same time so I can check both.

Try Toituer. It does have the ability to link more than one account. However, you will need the premium version to get that feature

Another alternative is to simply use two different twitter apps, one for each account. Not the most elegant solution to be sure.

Twidroid pro works for me, I know it's a paid app, but it's the best twitter app I've used so far...

Check out Seesmic over here http://www.droidforums.net/forum/rescue-squad-help/39815-tips-tricks-2.html#post447407. If I remember right, I think you can do that with this one. Check it out and see if it works for you.

Touituer or Hootsuite

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Seesmic is a great app... and its free!