|Fax to Email is only practical if you have a broadband connection. If you are going to send an Email on a dial up system you might jest as well send a fax.|
As the OP has a dial up connection then the original fax is most applicable.
But as wizard-Fred says, you can only use the Internet or fax separately, you cannot use them both simultaneously. However, if you have a V92 modem with Modem-on-hold, an incoming call, such as voice or fax, can put the modem on hold, attend to the incoming call and then resume the Internet session.
As far a security goes, conventional fax is probably the most secure communication system you will get as it is a point-to-point system and is very difficult to intercept.