I know we should have an intention when praying the Rosary or attending at Mass. But I feel very swamped with who I could offer the prayers/mass for that I end up relying on God to give the graces to whom He pleases, as He knows better than me and who needs them more.

Does offering these things give extra graces to the people whom it is offered for? If offering it for multiple people will the grace get split?

I do a combination of what Ladislaus said, plus stating specific intentions.  I figure Our Lady knows best in what proportion to distribute our prayers, but I also feel like I should name specific intentions too, as it seems to me God & Our Lady would want that.  [i.e. I think God wants us to pray for specific intentions too, even while we trust that He & Our Lady know best what is needed.]

At Mass, for example, I will ask Our Lady to do what she thinks best for my Mass & Communion intention, but then I will offer it for either several specified intentions, or maybe, e.g. if someone just died, for the repose of that person's soul.  "Does offering these things give extra graces to the people whom it is offered for? If offering it for multiple people will the grace get split?" On the one hand, the value of the Mass is infinite, so in a sense, I don't think it gets too diluted, so to speak.  But on the other hand, a Mass offered for one particular soul seems like it would be more valuable for that soul than one offered for dozens of people.  Plus I can be indecisive:  "Should I offer this Mass for intention X or intention Y?  Or, intention Z? etc."  So that's where I depend on Our Lady to distribute all for the best, in the best proportions, and I don't worry about it. 

And for Rosary intentions, I try to include big picture intentions, like for the rectification of the situation in the Church & the world, as well as more personal intentions, like for any conversions needed in my family, or the repose of the souls of family members & friends.  And even there, I might start with general & then "especially" particular:  e.g. "For the repose of the souls in Purgatory, especially Dad & Mom..."  And finish off with "for all those intentions I should be praying for" or "for all those who've asked me to pray for them", so as not to get too bogged down in listing off a million names, which I couldn't remember anyway.  But always knowing that Our Lady will sort it all out for the best. 

I have a lengthy list which I refer to for my intentions, too numerous to go through every time so I just mention my prayer list and leave the rest to Our Lord and Lady, they see the list!

Pretty much the same here, but always at or very near the top of my list is always, always, always: "All of the the holy suffering souls in purgatory, especially (here I name names.)"

Although I mention names, I also include even more and say "and all of my ancestors and relatives," and "all the clergy, clerics, nuns, popes and and hierarchy in purgatory especially those I may have known or who knew me." 

No matter what, Mass, rosary, or cleaning up a mess, I try to always to remember to include the holy souls in all of my intentions, especially at Mass, more especially at the Consecration.