You don't necessarily have to link against MySQL libraries in order to use MySQL in your application. All you have to do is to "speak" to a MySQL server using the MySQL protocol; in which case, you are connecting to the MySQL server as a client, i.e. as a user, and then you don't need to release your software under the GNU GPL license.
The question is, how can your program speak the MySQL protocol? One possibility is using the official MySQL client library (connector), which is GPL'd. If you do this, you are statically linking your program to a GPL'd library, and thus you need to release your software under the GPL.
Alternatively, you can link against a third party client library, with a different license. For example you can use the MariaDB Client Library, which is LGPL'd (and is thus compatible with proprietary software) and provides the same API of the original MySQL client library. See the MariaDB licensing FAQ, which also applies to MySQL, for more details and advice.
It may be tricky to distribute the MySQL server along with your software: you could simply require your customers to download and install MySQL by their own, or develop a simple downloader to be distributed with your software.