Grade tests, not homework. (Homework is drill for tests, is practice problems.)
Make a key before GIVING (not grading) the test.
Construct tests by using unassigned drill problems from the text. (Easy and relevant. If a kid does the extra drill and has it easier on the test, that's a feature, not a bug--encourages drill.)
I don't think a couple sections of 35 people is arduous grading. Look what HS teachers do. Just buckle up and do it. I don't think the size is sufficient to justify the work in making some system, nor the drop in question quality from going to multiple choice.
Let the pre-calc person do the pre-calc grading. Have him take the test and then construct a key that you review. After that, leave him be.
Comment about HS grading was not meant to be harsh. More calculational. Like if you told me the wife/husband had the family car and the job is 2 kilometers away, what to do (the answer is walk, walk faster, wear running shoes and keep dress shoes to change in office, etc.). But if it is across town, that is another story and we can brainstorm larger systems (bike, beater bike, carpool, bus, etc.)
Figure 16 week semester, MWF single periods. If you do a test every second week, that is 8 (7 if you skip last test), plus a final that is twice as long. Figure 5 essay questions per test, that is 35*5*2=350 questions per test. Figure 30 seconds per question, that is 120 questions per hour. (When you are flying, 30 seconds is achievable.) So three hours grading. Call it four, with adding totals and such. Double that for the final. So 7*4+8=36 hours of grading. Not great, but not end of the world. You could reduce it significantly by doing 3 interim tests plus a final (3*4+8=20 hours). It will be less effective coaching for the trainees, but will be more research time. They won't complain about less tests either...so that is a moral choice, but an easy expedience. [Of course, it also takes time to construct a test and a key. However, if you are using drill problems (or very close approximates), the test design is not too bad...key is a little more.]
Make the tests physically easy to grade and easy to take. No blue books. Each test is 5 "essay" questions with each question worth 10 points (easy percentages and addition!) You can do multiparter questions though. Use 1 (ONE) page per essay question. Make it single-sided (yes, kill the trees...paper is free, to you.) This will be easier for you to grade. And it gives kids the back if they are prolix. Use a standard format with a box in upper right for the partial score (and for front page the total score). Tell the kids to box their final answers (you can still grade on work done for partial credit/deductions but at least finding their final answer within the chicken-scratch is easy). Note your key should already say how much you expect for each subpart of the problem. The AP system is a good guide for how to do this.
I would grade the exams in order (question 1 for all, then question 2 for all). Think it will give you a little better speed and definitely better accuracy for the students. I would probably not separate the pages since you only have one grader (you) and since it becomes a nightmare to reassemble them (even with a numeric indicator). Keep a stapler to hand to deal with the inevitable exams that start separating.
Use your judgment on curving or not. Personally, I think if you can shoot for tests of reasonable difficulty level so that the standard 90-80-70-60 divisions work, that is better. But if you have some calamity, you might consider curving. (With multiple evaluation points, I would not sweat it too much as the students have a chance to recover.)