I had the same problems as you so I dyed the grout dark grey in my kitchen and dinning room and while it does look 100% better than it did I would not recommend you do this yourself.
If I could go back in time I wouldn't have done it and would have instead got some quotes from a professional on how much it'd cost to re-grout the tile with a darker grout and also got some quotes on how much it'd cost to just rip out the old tiles and grout and put in new tile with a darker grout and made my decision based on those quotes.