I am no longer satisfied with my current design wysiwyg program.
If I obtain Coffee Cup can I move my content into Coffee Cup?
If so, how?

Is your current site responsive?

If you mean, can you import it into any of the wysiwyg programs, that answer would be no. If you want to create a great responsive website using CoffeeCup software, then you should trial the Reponsive Site Designer program. Take it for a spin and give yourself plenty of time to get to know it.

Although it's not the type of wysiwyg software that you are used to, I think you will find it is extremely powerful and there is so much more you can do with it than most every other site builder out there. It's a little bit of a learning curve, but it's worth it and there are a lot of users here on the forums to help you as you learn and create your new website.

You don't have to do it all "from scratch" though.

Go to your live site and copy/paste sections of text into Notepad (or similar). This will strip the formatting so you can copy just the text into your CC app (HTML Editor, RLMP, RSD).
If you don't already have an Images folder for your site, create one and copy all your original image files into it. Now you have those ready to go.
If you have any external scripts (example: slideshow), copy/paste everything associated with it into a Notepad file so you can easily copy it into your new site files. This would be a good time to ditch old, outdated stuff. For example, if you have any Flash based items, replace them with newer Jquery versions instead.

If you're using RSD, it has "Resources", where you can add all your stuff (images folder, video files, scripts). There's also "Settings" which makes it easy to add scripts to your page or meta-tags to the head, for example. Use the HTML Element to add the actual item (form, map, slider, etc.)
As you build your page layout, you can select an element – paragraph, heading, etc. then go into edit mode and paste your previously saved text into it.

Grab a free trial of RSD and give it a go.