I use If This Than That to do stuff like this. It's like glue for internet services.
There's a bunch of recipes that post Facebook wall updates to Twitter automatically for you. Here's one: http://ifttt.com/recipes/1854
This would save you from having to open a Twitter client. But now you've got posts in Facebook truncated in Twitter because of the length restrictions.
Why not just switch to a better Twitter client? One that has TweetMarker support?
If you want to try the best-of-the-best but bleeding edge check out the currently-in-beta Tweetbot for OS X. It's a sweet, sweet client and with TweetMarker support you can open it, tweet a new tweet, and then close it down and it doesn't lose your position. It even syncs the position other TweetMarker-enabled clients.
If you're not willing to run beta software try Twitterrific -- also has TweetMarker support and also doesn't lose your position if you open, tweet, and then close it down again.
If you use one of these Twitter clients and combine them with an IFTTT recipe that reposts and tweet you make with the
#fb hashtag to Facebook you'll be all set. Here's the recipe I use.