I have a dependent variable made up of 3 categories and 14 binary predictor variables.
I have tried using
nnet/multinom packages in R.
Is there a better approach than multinomial logistic regression for this particular scenario?
Given your description of the situation, you are using the right model. There is no problem with having discrete IVs with multinomial logistic regression; MLR does not make any assumptions about the nature or distribution of the IVs. However, I wonder if your IVs are not orthogonal. It's hard to tell, but you may be describing some effects of multicollinearity.
I'm not sure what happened with R, you would need to show your code and data and perhaps the error messages for someone to help you figure that out. Questions about those sorts of issues should be asked on Stack Overflow or the R-help mailing list, though; they are off topic here.