My question is this... if two people in a relationship sign an apartment lease, and things don't work out, can one party break the lease? And if so, is the other party that stays responsible for the full rent amount or is the "breaker" still liable for what he/she signed for? This is for Washington state.

It depends on how the lease is written. If it has paragraph saying that each party who signs is jointly and severally liable (and most leases do have that paragraph), then the person remaining in the apartment is responsible for the full amount of the rent. But that person can sue the person who broke the lease for their share of the rent.

So the "breaker" is not necessarily liable to the landlord but they are liable to the remaining tenant.