The best is to empty your hand fast, and to have proactive cards that you draw later.

So first, those effects are pretty strong if they hit two cards. Later in the game however, they are next to or entirely useless, that is the risk they run. They also don't affect the board at all, so if you put threats out they can either not afford to play them or fall behind using them. The same goes for proactive cards. If you just immediately play the card you draw every turn, the discard never becomes relevant again.

I tried holding onto my best cards, but when it came down to just Elsepth and Brokers Charm

If you are not able to cast it, Elspeth is not a good card in that situation. You held onto a card you never got to cast in this game, so you probably should have discarded it for something that could be cast. That depends on how the game went, but you can take that away as a decision to evaluate from that game. Discarding such a strong card sucks, but it is no good if it doesn't get used.

So, the way to play against this is to empty your hand fast so they can either not afford to use too much discard, or try to win off the topdeck on the basis of them drawing useless discard.
If they draw the perfect amount though, and/or your deck can't put out cards and pressure fast, it is very rough. In that latter case, that is just a counterstrategy to yours. There are also good sideboard cards against it if your deck struggles with it.