For já, I’ll refer to other topics about it (one of them is yours, actually ):
As for mesmo, in this context it means really.
- Precisava mesmo de ficar boa = I really needed to get better.
In other contexts, it can mean same, even or for real.
- Este é o mesmo carro que eu comprei. = This is the same car that I bought.
- Eu ainda sou a mesma pessoa. = I am still the same person.
- Liga-me, mesmo que seja tarde. = Call me, even if it’s late.
- Mesmo que tu não gostes, tens de comer a sopa. = Even if you don’t like it, you have to eat the soup.
- Pessoa 1: Eles são loucos! Pessoa 2: Mesmo! = Person 1: They’re crazy! Person 2: For real!