The directories get searched in the order they appear in LD_LIBRARY_PATH.
Keep in mind that the loader also looks elsewhere, not just LD_LIBRARY_PATH, which may or may not be a factor.
Since you asked for a "better solution", here are two:
Get this broken shared library built correctly
There is a well-established mechanism and convention for versioning shared libraries so that different versions of the same shared library get loaded. A program that needs a particular version of a shared library will load the one that it needs.
In fact, it's a safe bet that you already have a bunch of different versions of multiple shared libraries installed, due to applications in your Linux distribution that need different versions of the same shared library.
This is nothing more complicated than not using the same name for incompatible versions of the same shared library. There is a well-defined convention for naming actual shared library filenames, that work together with the linker in order to make this happen (the -soname link option, see your linker documentation for more information).
Don't use LD_LIBRARY_PATH
It is possible to link an executable and embed in the executable itself a pathname to search, first and foremost, for any shared libraries, either before or after LD_LIBRARY_PATH.
Remove all directories from LD_LIBRARY_PATH.
Use the -rpath, with/without --enable-new-dtags or --disable-new-dtags option when linking your executable. The correct set of options depends on your specific details, and specific versions of your linker. See your linker documentation for more information. You mentioned you tried to find information on this in Google, but all that Google will do is, perhaps, refer you to the same documentation that you already have: the manual pages for the linker. That's the best source for complete information on using this or any other linker option.
The best solution depends on your specific circumstances; whether you're building the executable, and/or shared libraries, how easy/hard it is to change whatever you're building; or whether you're not building anything at all, just black boxes that need to be executed.
