Even cheap spectrum analyzers are still going to be well out of that price range. You *could* buy one of those $20SDR dongles and turn it into a spectrum analyzer, but the sweep rate will be miserably slow which is not good for catching transients never mind direction finding. The reality is that anything resembling professional equipment for your task will cost $1k (old, used, limited) to $10k (entry level lab/industrial stuff). That doesn't mean you should give up, just realize that you'll either need to schmooze someone who does have the equipment or hack together something that would give most folks over in the Metrology subforum a severe case of the "tut, tut"s punctuated by fainting spells.
Assuming you want to do the latter, my suggestion would be to stick in the time domain for your first pass at hunting down the noise (we're taking advantage of the fact that you suspect it's intermittent here). Buy a cheese-grade ($50-$150) portable oscilloscope; first make sure it supports "Normal" in addition to "Auto" sweeps, then maximize bandwidth subject to price constraints. To detect E field noise, stick a wire in the socket and let the other end dangle. To detect B field noise, stick a wire in the socket and connect the other end to ground (not earth ground, the pocket scope's ground). You will want to research the types of sockets on these scopes to either pick one that facilitates these wire attachments or to buy an adapter. Read up on dipole and loop antennas to understand the limitations of this approach, but if the interference is bad enough you'll see the wiggles it leaves behind. If you don't, it might be conducted emissions. Grab a 10X probe and look for spikes on the mains. Preferably a fixed 10X probe because cheese-grade "scopes" don't support mains voltage at 1X and even accidental momentary exposure to mains at 1X could fry them and you. Hell, I'd probably wear gloves while handling the thing. But between those 3 techniques I'd give you slightly better than even odds of finding the problem.
The less-fun but more likely to work approach is to schmooze someone with the proper gear. It's easier than you think if you find someone who isn't busy at the moment (which is the real trick). People with expensive equipment and hard-won expertise like to use it. Just approach them when they are 1. not busy, and 2. hungry (you *did* remember to bring a food bribe, yes?) and it'll work 80% of the time. Honestly I'd suggest starting with this approach but since you're in grad school I would guess that you want to take a swing at it yourself first :-)
EDIT: careful about focusing too much on the transformer; I've got a cheap wall-wart that I bet could give your NMR machine the fits from twice as far away.