Can El Niño/La Niña occur again after normal state?
If you look at the ENSO index record, you can see many instances of El Niño or La Niña occurring multiple times in a row. For instance, the period 1977-1983 and even after that is a succession of El Niño events with normal conditions in between and no La Niña period.

Multivariate ENSO Index (MEI) from NOAA