How Can I Get My Hanging Fuschias To Continue To Blossom?

You should consider replacing them. Maybe they are not getting proper sunlight and air. So you should change the place and hang them somewhere, they get what they require.

Fuschias are water guzzlers so I find that fertilizing them every week promotes continued blooming. Using a standard plant food such as Miracle Grow for flowers works well.  Also making sure that you rid the plant of dead blooms/pods will help.  I know that fuschias drop most spent blooms on their own but it will help if you remove all of the spent blooms and any pod that may be hanging from the branch.