You may want to 'normalise' any region parameters:MIDI:Region Parameters
If you have the notation looking right in the Score Editor, you should note that generally this does not alter the MIDI note events, explaining the 'mess' you get in Sibelius. Try using Score:Functions:Quantisation:Fix displayed note positions and durations.
"What am I doing wrong?"
Using Sibelius? :-)
If you do what CCTM suggested and also possibly consider using "Sustain Pedal to Note Length" you should be fine.
Also: To check what your results would look like without Logic's rather smart "Interpretation Mode" and "No Overlap" features in score, you can turn them off before you export your MIDI tracks.