Shaving the bottom is way easier if all the strings are the proper height in relation to each-other....Tuning the slots works well when there are individual string height issues....but to tune the slots you need some sort of nut slotting files which are not cheap. There are folks who have used tip cleaners and such and one can make it work...but the real thing makes a big difference in keeping everything in proper alignment, especially if you are a beginner.
If you are unsure of what you are doing and concerned you might mess it up, you could order a nut blank (or two as they are cheap) make it like the one you are fine tuning and then adjust as you see fit....This way you can always just pop the original back in if you should mess up..... Otherwise just give it a go and hope for the best...
Either way you do it just take your time and be patient....make a little change and string it back up....repeat until you like it....Where most folks mess up is they try to "save time" by getting the job done in one shot and go too far (by removing too much) just because they do not want to keep slacking and tuning....you really only have to slacken the strings enough to get them spread out of the way (without much tension preferably) so I take my string winder and go about say 20 turns (if you count your turns whilst slacking you can get close to pitch when re-tuning just by counting, then fine tune without being too far off)...You do want to check your string height(s) with the strings up to pitch by the way...
So in a quick step by step..... slacken, remove nut, sand on flat surface ....**make sure you are sanding evenly so that you are not taking more off one end than the other...two ways to accomplish this are to use a guide of some sort or use little pressure and rotate your nut every so many strokes, as we tend sand with uneven pressure...Also I would suggest something like 180-220 grit as you do not want to go too fast**....place back in position, re-tune, check height and repeat until you like it...Then put one small dot of medium viscosity super glue in the middle of nut on the side face that mates with the fingerboards end grain and hold in place until it sets....if you put (or get) the glue on the bottom of the nut it is likely to take wood with it when you want to remove it again in the future!
However, with all that said the other option is to take it to a luthier / repair tech and have the job done....
Hope this helps!