Is there an iPhone app that lets you post to Twitter and Facebook Pages as yourself?
I let Twitter do it automatically but I'm not sure it will carry your name but rather post as the page. You'll have to try it and see. It may be that it's functionality not included in Facebook itself. 

Here is the instructions from Twitter gel pages:

To connect your Twitter account and your Facebook profile:
Log in to the Twitter account you want to associate with your Facebook profile.
Go to your Settings menu's Apps tab. 
Click Connect to Facebook.
If you are not already logged in to Facebook, then you will be prompted to enter your Facebook login credentials. Enter your credentials and click Log in to sign in to Facebook.
You will then receive a prompt explaining that Twitter will receive certain information from your Facebook account. In order to continue, you must select “Okay.”
You will then be prompted to select the privacy settings for who will see your Tweets and Retweets posted to your Facebook wall. It is set to friends by default.
Click “Okay” to complete the process.
Your Tweets and Retweets will now post to your profile Facebook wall and your username will be displayed there as well. @Replies will not be posted.

Note: To link your Facebook updates to Twitter, use this application on Facebook: https://www.facebook.com/twitter.

To connect your Twitter account to a Facebook page:
If you have a Facebook Fan Page, or are the admin of one, you can post Tweets and Retweets from your Twitter account to your Facebook Page.

Follow the steps above to connect to your Facebook profile.
If you have not already done so, visit your Apps settings and grant Twitter the “manage pages” permission for your Facebook profile.
In your Apps settings, Select the page you'd like to connect to.
When prompted, click to allow permission to post to the selected Facebook page.