Practical addition: your hands falling apart when both are busy may partly be a counting problem rather than a reading one.
A lot of adult beginners read pitch fluently and rhythm approximately, and get away with it one hand at a time because there is nothing to line up against. Two hands exposes it instantly, and it feels like a reading failure because that is when it appears.
Worth testing separately: take a line, forget the piano entirely, and clap the rhythm while counting out loud. If that is shaky, that is your two-hand problem and no amount of note reading will fix it.