You can definitely have multiple values on one of the axis in the dual-axis chart, but not both (without hacking).
so whats the hack?
One way to hack it would be to unpivot the data so that measures/values are not separate columns - instead they are in the same column, but on different rows, while another column has measure names. So, instead of
State Sales Profit Budget
Texas 100 35 120
Alaska 200 23 150
you will have
State Measure Value
Texas Sales 100
Texas Profit 35
You can then create two calculated fields that pick values you want, and place those calculated fields one on each axis, e.g. to only pick Sales and Profit, use
IF [Measure] = 'Sales' THEN [Value]
ELSEIF [Measure] = 'Profit' THEN [Value]
You may need to the hide all nulls, and maybe do some other tweaks, but this should work.
Tableau has a free Excel addin to unpivot data: http://www.tableausoftware.com/about/blog/2012/03/reshaping-data-made-easy-16353