I was a flex employee when I first started at the hospital. As long as I met the minimum requirements for shifts (4 shifts a pay period, with at least half of those weekend shifts), I could put in for as many shifts as I was available for. I had no problem making full-time hours: I put in for 4 shifts a week and got all 4. Check with your staffing/HR department to see if you can put in for extra shifts...in my experience, most places will let you.
I never had to be "on-call" like you describe, but nearly every day I'd get a call from staffing asking if I was willing to work that day/night or the next day. So it'd be up to me if I was willing to go in.