Correct. When I was running a fax application I did the same as you are
doing now - disconnected my broadband modem to connect the fax modem. Send
fax and then switch back.
I never tried with both modems connected at the same time. Theoretically it
should work out with traffic routing to the correct modem as needed.
However I was worried about the connection for the internet getting messed
up so did it the hard way to avoid a potential problem.
Also have never tried any of these but there are online fax services. Most
charge a small fee for sending faxes on your behalf and receiving them and
delivering them to you via email. A few of these have a "free" level of
service that allow so many faxes for free a month or free with other
restrictions. Using google, I found a discussion forum that has info (and
discussion) on these services: http://www.faxbeep.com/internet-fax-service/