I play a spell; my opponent reacts with Lilting Lullaby targeting my spell. I react with Abandon targeting my own spell. After everything resolves, can I still play more spells this turn?

Yes. Lilting Lullaby mistargets and instructions tied to the abandoned spell are ignored. The linked "controller can't play spells" instruction does not apply if the counter instruction is ignored for any reason (359.3.e.14.a).