Well dude, you're through the worst of it. I wore the same type of sling for four weeks(the one with the big, fat pad and a waist strap)after having a labrral repair back in September. I kept myself busy doing odd chores around the house. I spring cleaned one handed (although that aggravated the other arm––Doh!), sorted through several thousand rounds of brass, went through all my survival/ALICE/camping gear, and inventoried all my ammo (and all my dad's, too). I put 3,000 rounds of .223 on strippers, detailed the interiors of two cars (albeit very slowly) and did a lot of online research on food storage. Not to mention, I amassed a prodigious post count on ARFCOM (1,500+ posts in six months of being out of work).
Actually, if not for having this site, I would've went absolutely crazy. I also bought a few new guns, tinkered with several ARs, and stocked up on mags and accessories.
I cheated while I was in the sling. First, after the first week, I took it off before I got in the shower, and would leave it off for an hour or so while I was getting dressed. The doc gave me a list of exercises to do at home, and I pushed them to the limit. I would also remove the shoulder strap whenever I was sitting and had support on the forearm, although that may not be advisable in your case. While getting dressed, I tweaked the shoulder a few times, and that bitch throbbed all day. By the fourth week, I would wake up sleeping on my stomach, on top of the arm, and the arm would be popped out of the sling. So I just started sleeping without the shoulder strap on.
I was supposed to wear the sling for six to eight weeks. After four full weeks, the doc said I was ready to get out of it permanently and begin physical therapy––that's where the real pain begins! After 10 weeks of intensive PT, I was able to return to full capacity with no restrictions. I was doing pullups again after 8 weeks in PT. It's now been five months since surgery and the only thing I can't do are palms-forward pull-ups. Dips, bench, shoulder press (that hurts a bit, but doable), and chin-ups are all possible. I still have some discrepancies muscle-size-wise on the left side, but strength is 75%+ what is was prior to surgery. My only range of motion limitation is reaching behind me, like when grabbing the seat belt from a car's B pillar. I don't even notice it.
Here's to wishing you a full recovery!