You might be able to use your moving days to get the time off. I don't know how far you "live" as compared to where you got based. You have 1 packing day and 1 unpacking day plus 1 additional day for each 350 miles. With some creativity, you might be able to sqeak a stretch of 5 or 6 in a row depending on your reserve schedule or green line.

Moving ahead, the best deal in town is to bid a highspeed line and burn a sick call for a stretch of 4. Highspeed lines are usually always 4 on, 2 or 3 off. One sick call will take out a stretch of 4 highspeeds and you only burn 16 hours of sick time. So by calling in sick on a highspeed line, you are guaranteed to have 8 days off in a row. A little trick to put in your toolbox for later use!