Yes, they wait in line. That is standard in virtually every correctional facility, and Lea County is no different.
Shower access is not open all day. Facilities run on schedules, and shower time is a designated window, not an on-demand amenity. The number of shower heads is limited relative to the population, so inmates have to plan around that window and wait their turn when it comes. If someone is slow or the timing is off, they may miss their slot entirely.
It is one of those unglamorous realities of incarceration that does not get talked about much but adds up over time. Privacy is minimal, the water temperature is inconsistent, and the whole process is rushed. Learning to work within the schedule quickly becomes second nature, but the first few weeks can be an adjustment.
If your person is heading there or already inside, the advice is simple: know the shower schedule, get there early, and do not expect much in the way of comfort or privacy.