It sounds to me like you are lacking guard digits. These are the extra digits (or bits when thinking in binary) that preserve the accuracy of your intermediate variables. If you are using all 16 bits then you don't have any guard bits available, and whether you use a divide instruction or a shift instruction doesn't affect the result.

Adding 0.5 lsbs before rounding changes your code from biased rounding to unbiased rounding, but you need somewher to store that 0.5 lsb.