Without using dowels, how do I join 2x4s edge to edge to be 1.5" x 7"?
I would definitely use Pocket-hole joinery

My brother got me a jig that is very easy to use, and the joints are incredibly strong even without wood glue; also if you use glue the joins are "self clamping".

Basically this allows you to edge-join, or do 45 degree angles, etc. just by drilling a couple of perfectly angled holes and screwing the pieces together such that the screw doesn't protrude from your work.

Obviously, I wouldn't get a jig for just one project, but I have totally stopped using dowels, biscuits, and other fancy techniques in favor of pocket holes.

Hope this helps!