Decide on the conversion then look for software which does.
get my texture to map correctly to a sphere without stretching
as you can read from previous - miracles do not happen (or, such mapping has not happened to be developed yet).
As to what you can do to correct image which will be distorted - there is a way to turn plane into sphere fast by using Warp tool:
Warp once 180 deg so that you get map rolled into half cylinder, then warp so that half cylinder turns into sphere.
If you had imported your map as a plane (using addon) all you need to do is:
subdivide once so that you get 2 square faces then subdivide all say 12 to 16 times
Ortho view look at the end of imported plane, place 3d cursor 2BU below the middle of the plane and Space key search ‘warp’. Enter 180deg.
rotate all Z 90 and repeat warp; this time enter 360deg
remove doubles.
If you do not push continents close to poles distortions will be less visible. You can use Texture paint to correct texture at the poles (use Stencil paint option or Clone paint. Do not forget to save painted image from UV editor after the painting is done.
i make a lot of maps for Celestia ( space sim) and other programs
normally it is best to start with a image that IS IN "Simple cylindrical " format
from looking at your image this is not even in cylindrical
– the left and right sides DO NOT match
– the 180 lon and +180 long are THE SAME and need to match
the left and right side do not match
for some really BAFFLING reason most so called “guides” have people make NON MAPPING square maps ??? mind boggling baffling !!
but a few things can be done to get you close
you can assume that this is a Mercator projection that is 60 north to 60 south
remap it to 90n to 90 s in Simple cylindrical ( some times called equal area rectangular )
and then fix the left and right and inpaint the poles