When I mentioned jobs in previous posts, I was referring to jobs outside the industry. If you're able to get your foot in the door while in grad school, jump on the opportunity. If you're thinking about working 40 hours a week at Starbucks while going to school, I'd advise against that. That was more of my point in the previous post.
Otherwise, an internship is less work than a job, so that won't take up all your time. You'll be working maybe three days a week. If you're talking about entry level paid work however, it will be very time consuming. As a PA, you'll usually need to show up on set first and be the last to leave, so that can easily turn into a 14 or 15 hour day after a commute. As an executive assistant, free time will also be pretty sparse.
You might be better served trying to do freelance PA and script coverage work. That way, you'll be reading a lot of scripts, which is really important for writers to do, but you'll also be able to set your own hours. If you're able to get steady freelance PA gigs a couple days a week and do coverage, you can probably make just enough money to get by, but also have the time to write. The only issue is that you'll need experience doing both those things in order to get paying gigs, so you'll need to start out as an intern first.
Regarding internships, I'd recommend getting one that requires daily coverage, so that you can get used to it and get good at it, so you can eventually get paid for it. This industry can be incredibly hard to break into because of how time consuming it can be. You can work an entry level job and have no time to write new scripts or develop projects as a producer, so you end up just working your way up from the bottom, which is one way to make it in this business. Another way is to write a lot of stuff, network, and get your work into the right hands. With some talent, a little luck, and a lot of perseverance, you could end up optioning and/or selling something and moving up really quickly.
Any entry level job though is going to be very time consuming because, in one way or another, you're supporting someone else. You're either a PA supporting development or production in one way or another or an assistant supporting an executive. There's really no way to just jump up the ranks, and an MFA won't really help with that. It can make you stand out amongst the other PAs or assistants, but no producer is going to hire you for a writing assignment or to show run a drama series because you have an MFA.
Now, I say that, but there are exceptions. If you spend your time writing, and you have the right script that gets to the right agent, manager, or producer at the right time, you could end up in a meeting and get a job as a staff writer on a series or possibly even get a writing assignment. Though, I still think it all boils down to networking, which was my initial point of the post.