Alright gang, let's see if I can shed some Target HR light for you.
First of all, swap shifts are NOT required to be done at all. Target gives each store the option of either doing it or not. It's a "privilege" to offer swaps not a right.
Secondly, you are able to pick up shifts any day, any time, as long as you a) know the work center b) won't go over 40hrs in a week c) aren't a minor and violating your work permit. So you can pick up shifts on your unavailable days.
Finally, if you are scheduled on an unavailable day, unavailable time, or date that was requested off you need to let your leader or HR know right away (the schedule does go up 2 weeks early). MyTime has stopgaps in place to prevent this from happening, but I've seen it happen about a dozen times since MyTime rolled out.
If you let them know early they can find someone to take that shift. If you wait until say the day before, they're going to be upset and may tell you that you're responsible for the shift still. Ultimately, you own your schedule and own ensuring that it's correct.
If you have any specific questions or want to run other things past someone feel free to pm me.