Your bike should work just fine. Your fitness will have a bigger impact on getting up that hill than the bike will. You have low enough gears.

If you want a new bike you can get one, but its not necessary.

Have you had a professional fitting? If you haven't you may want to consider it. Sometimes even small adjustments can lead to more comfort and better performance. I know when I got one my new position made climbing hills easier for me.

The bike deals you mention sound pretty good as long as the new bike fits well. If the frame is the wrong size or doesnt fit well its not worth it regardless of the price.