First of all, I happen to think much of the seatwork is junk and don't do it. lol. I do have them complete the Arithmetic, Language and Phonics workbook sheets, and sometimes a tablet page.
I have to do more grading than ABeka requires, so I have to grade some of the seatwork.
So here is what I do: They complete all their assignments in the book, and do no tear them out so that I don't end up with mountains of seatwork. Since I omit much of the tablet work, i don't have a huge problem with students not finishing. If they don't finish I write/stamp "Please Finish" across the top and send it home. (In the book)
On Wednesdays, I have them tear everything they do out and I grade it for a classwork grade.
As far as looking everything over, I walk around the room in between my reading groups and spot check for general problems. (trouble with marking vowels, counting backwards, and odd/even numbers are my issues right now for example

) Sometimes I circle incorrect answers and have them redo some of them. I also often check to see if they are doing the correctly and will send it home to be redone if I'm not happy. ABeka does not require you to grade or ever LOOK at any of it. But this system works for me.
I know many teachers that have a folder for "Ongoing" work that is incomplete or needs to be finished, but I know some of my kids would end up with TONS of papers. So instead I send it home and after missing some after school time for wasting time in class, and if your parents back you up on this it often helps. I have one student who doesn't get to go to Tae Kwon Do if he wastes classtime, so this method fixed the time wasting FAST.
Oh and I also only send home GRADED papers and keep the rest in the book.
I'm no expert but I'm glad to share what works for me!

Sorry so long!