Your question is confusing and seems contradictory.
Either you want to work with an external MySQL server, and that means that your application opens a (tcp(7) socket) connection to some remote machine running mysqld. Then you have to use libmysqlclient (because that is the only library to run such a connection and dealing with the MySQL client/server protocol on the client side).
If that mysqld server is running as a different process on the same embedded Linux system you should have some way to start it (probably as part of the init scripts on it). Then you still use a socket communication to it, and you still need libmysqlclient. A possible difference with a remote machine running mysqld might be (but I am not sure) the socket family. Perhaps libmysqlclient is using unix(7) sockets in the special case of connecting to a server on the same machine.
Or you don't want any external server. You might consider libmysqld but as Romain answered it is deprecated and is becoming unsupported (so I feel that would be a very bad choice). Then all the database code is running on your embedded Linux computer, which also has all the data storage. In that case (relational database & data & storage on the same embedded Linux computer), I would recommend using sqlite instead because it is well supported and quite stable.
If your mysqld daemon is running on a remote machine you cannot (realistically) avoid libmysqlclient (otherwise you'll need to rewrite most of it).
libmysqld(which won't be supported anymore)? You should explain that. Or why can't you start amysqldprocess in your embedded Linux system?libmysqlclient; it is the only library knowing the protocol to connect to the externalmysqld. In other words, you should edit your question to explain what do you mean by "external MySQL server" (sincelibmysqlddon't use any external server)