OP, I hear you. Toddlers can be sooooooo trying.
Now, I've been told off for being sanctimonious on another thread, so I will preface this with a disclaimer that I'm really, really not trying to make you feel bad. You are obviously a very thoughtful parent who is doing her very best under trying circumstances (you have two children. This is very very trying!!).
But please, for your sake, try to cut the humiliation out of the picture. Honestly, no one else will even notice if your kid doesn't make it into the class on time. He is 4. He is not the first and he won't be the last child to have an ill-timed tanty. Literally everyone there (assuming it is a child's class!) has seen it all before and if they are thinking about you at all, they are thinking either "oh, poor HadEnough" or "thank fck it's not my kid this time".
Allowing yourself to get worked up about what other people are thinking is only laying pressure on top of an already-annoying scenario. I know it's hard, but do try to let go of it.
I was in a baby singing class once with a mother who was hissing at her son to participate, chastising him for not wanting to get up (despite the fact that loads of the other kids were sitting on their parent's laps, too) and threatening him with going home. I know this isn't you, OP, you seem very thoughtful. But it really is a bad look - she was so clearly embarrassed and taking it out on him, when actually nobody else gave a sh*t.
If your kid is smashing something up in the corner and running around pushing other kids over - by all means give them a telling-off; that really is quite mortifying. But a kid who's being clingy and taking a bit too long - I can understand why that would leave you exasperated and flustered, but humiliated? That's on you.
Lots of good advice on here, hope things improve for you OP, and that you can get a break too. It really is so bloody hard! 