Need Help

Hi everyone,
First time posting and desperately need assistance with finding the right lens for my girlfriend’s Nikon N50 with a 35-80 zoom…your feedback would be greatly appreciated, as I am not a camera expert. Obviously, she’s been really limited with what she has now. So, I am hoping to find something that will help her zoom in on objects in the distance… compatible for indoors and outdoors shooting… Based on my research thus far, I’ve been kind of leaning towards the Nikon 70-300mm f/4-5.6D ED AF Zoom Nikkor or the SIGMA 70-300mm F4-5.6 APO DG Macro. Which one of these lenses is better, if any? I have budgeted about $250-$350, so please let me know if you have any suggestions. Thanks in advance for all your time and expertise.

Serge