Here's a quote from the Licensing section of the GNU website:
It has always been the FSF's position
that dynamically linking applications
to libraries creates a single work
derived from both the library code and
the application code. The GPL requires
that all derivative works be licensed
under the GPL, an effect which can be
described as “hereditary.” So, if an
application links to a library
licensed under the GPL, the
application too must be licensed under
(the specific page is discussing Java and LGPL, but the general point applies here)
If you want to license your original application as BSD, you must first create it without using that library at all.
Then, you can create a derivative work by combining your BSD application and the AGPL3 library, with the whole thing licensed as AGPL3.
This would effectively be a new, different application, and is not dual-licensing.
Since you want your BSD application to be used commercially, you would need to release it separately from the extended AGPL3 version of the application - which means two distinct links on your download page, with a clear explanation on the difference.
(NOTE: Anywhere I use "BSD" in this answer, it's important to note this refers to the "modified BSD" not the original BSD.)
What happens if I use other libraries which are licensed under BSD or anything else?
That's fine. So long as the license allows for the code to be re-released under a compatible license and the AGPLv3 is considered compatible (which the BSD is - see a full list here), then you've got no problem.
If you plan on using lots of libraries/etc then it's a good idea to maintain a distinct credits file, where you can list each specific component you use, its website address, licensing requirements, and so on.
(usual disclaimer: I'm not a lawyer, follow at own risk, etc)