24x7 shift schedules

Works just fine.

A couple of considerations, which may be what you are really looking for.

12-hour shifts, 4 teams, rotating through a 2 on, 2 off, 3 on, 3 off day rotation works well. The trick is to set up the legal normal work week in conjunction with rotation so the worked hours in any given work week are under 40. Employees work 7 out of 14 days, and the work week needs to be stated so that they have 3 full shifts plus one half shift each week to avoid OT. The 12-hour day is broken up by lunch and 2 breaks, which should keep you under 40 hours. Check with your local labor commission on what is legal.

A 6 to 6 shift is popular with employees, but it doesn't match the management team 8 to 5.

I recommend a 12 to 12 so workers have contact with management team on a regular basis. Helps group cohesion. It also reduces friction over day or night shift. Avoids having to flop teams from mostly day to mostly night on a scheduled basis.

If adding a swing shift during normal business hours, consider paying a shift premium for the 12 hour shifts.

Regarding employees who doze off on 12 hour shifts when there are no calls and no alarms - be realistic. Have scheduled maintenance routines to perform so they aren't bored to sleep. A 45 minute nap or work break will revitalize that worker. Any less does little good, much longer and you might as well let them punch out for the day.

Time off for personal business should not occur. All workers have plenty of non-work hours during regular business hours to handle personal matters.

Cross training is a better approach than OT to cover sick workers and vacations. However, if you are covering primarily calls you need to staff to cover peek with one person out.