Possibly not, since an unshared worksheet will always create its own (unshared) connection, and one wouldn't expect to be able to switch to something that hasn't been created yet. Also, an unshared worksheet only lists its own connection in the drop-down list of connections at the right-hand side of the toolbar, so there is no way to switch to a different unshared connection or to any existing shared connection, whether already open or not.
In the meantime, to get the content of an existing worksheet into a newly opened unshared worksheet, just use Ctrl-A, Ctrl-C (on existing), and Ctrl-V (on unshared). Not too hard.
I believe the OP wants to go from a Shared Worksheet and turn it into an Unshared Worksheet.
You could always contact Oracle Support and create an Enhancement Request.
There is a place you can place the request (along with other people's request/suggestion)
IIRC - I believe ThatJeffSmith said "no" when I asked on this forum. (I forgot the circumstances)
i like the idea, it's just the way worksheets, connections, and the framework works, this would require more work than we can afford to expend
These types of things are always in our minds for things to improve upon, and look at when thinking about what the next iteration of SQL Developer will look like, and how to build it to accommodate stuff like this much easier.