Nobody on here can diagnose, including me, but I will do my best to offer some advice.
Firstly, its very common for anxiety to manifest as a lump in the throat. So yes, its a fairly common anxiety symptom. Secondly, the "having" to do something to relieve it, is a little bit on the OCD side. Thirdly, I am going to save you some time, and a lot of frustration, and hopefully help you nip this in the bud, if possible.
Your compensatory behaviour (having to look down) is actually what is exacerbating the situation. Your brain is getting caught up in a loop. Its mostly unconscious.. it goes something like this:
1. Notice throat sensation (very very common, if i think about my throat here, I will notice this)
2. Worry that people will notice you swallowing (less common, but i can totally understand why this might happen)
3. Compensatory behaviour to avoid the catastrophic reaction your mind has come up with for if people see you swallowing (think about this btw, what do you fear?)
4. The catastrophe doesn't happen!!! the compensatory safety behaviour worked! Nobody reacted.
5. Better make sure to do it again next time to prevent the bad outcome. (go back to 1).
Here is what will make the situation worse.. thinking that you need to "treat" the throat sensation.. thinking you have abnormal levels of social anxiety... thinking you might have ocd... going to the doctor and getting prescribed an SSRI, or have them not understand. Feeling ashamed in any way. Feeling you are abnormal, or broken, or anything else.
Here is what will make the situation better.. remove the safety behaviour, the need to look down to swallow. You need to not do this, and then let your unconscious learn the safety behaviour isn't actually doing anything.
Two ways to go about this.
1. literally time yourself with a stopwatch to see how long you take before you need to swallow. If its really 10 seconds, day 1, push yourself to do 12 seconds. You can absolutely last another 2 seconds, because you already do 10.
2. next day, 14 seconds
3. next day 16 seconds
4. next day 18 seconds
5. next day 20 seconds
congratulations, your problem is now half as bad..
6. next day 25 seconds (it should get easier)
7. next day 30 seconds
8. next day 35.... you get the idea
Its going to be uncomfortable, but you can do it, and take your time, do it gradually, the gradual approach is the key here, if you do too much you wont get there. Slow, slow slow.
Altternatively, it might be worth visualising the situation and trying to recreate the sensation now, when you arent in class (its worth doing this anyway tbh), and practice lasting longer. Really try to *feel* the sensations and invite the uncomfortable sensation when you aren't there. Then time it and see how it goes. This should be safe, so you should be able to do longer, but again, if you struggle, increase as per above.
What you can do is visualise the hell out of it, time it, and go for 5-10 more seconds than your daily goal in class, do this in the am, then achieve your goal in class.
The solution is to face up to the problem now, while its relatively minor, trust me. This is called exposure and response prevention, its a part of CBT, and its your best option to get the job done.
Go slowly, slowly, slowly.
If it doesn't work for you, type in IAPT provider and your area in the UK, and find somewhere that can offer CBT therapy, it should be quick (ish), but you might as well have a bash at this on your own.
Above all. More important than anything, be very very kind to yourself. It's extremely easy to get caught in these kinds of loops. Don't label yourself as socially anxious (please, it will only make things worse), just treat it as an artefact of having a human brain (it is). Our brains pull all kinds of unexpected things on us, its okay, its a loop.
If you struggle with the speed, slow it down. All you need to do here is very very slowly push back in the other direction, do this, and you will prevent the safety behaviour, your mind will catch up, and it will become manageable.
Be kind to yourself, and good luck. You can do it.