"I feel I have become resentful towards these men for putting themselves first"
That is because you might have been socialised to put yourself last and they might have been socilaised to put themselves first.
Are you responsible for their work/supervisor etc? If so, how can they say no? If not, then tell the person who told you to tell them, that they said no. IYSWIM. (and their supervisor if that is a different person).
Are you worried about getting them into trouble? if so, why?
"...for refusing to help with situations that THEY have created..." Who can you tell that they have cocked up?
"...for forcing me to effectively clear up after their incompetence..." Are they forcing you or do they just not care if it fails? If they do not care, should they be in those jobs?
"...and for not helping when I KNOW they have the capacity to do so."
But it's not really 'helping' if it was their job in the first place.
I think that you need to take your kids and their visibility, and your biological sex, out of the equation, I do think these situations are probably sexist, but you need to take that out of the equation too, IMHO.
They have stuffed up so how will their boss make them make it right and if they cannot make it right, who else can - someone who is not you!
"It just feels like I'm wobbling around at the top of a ladder asking for help and they're having coffee looking up my skirt and smirking while I sweat."
It feels like that because maybe that is actually how it is. By looking up your skirt do you perceive a threat from them or do you mean they are laughing at your situation?
I think you need to put that in professional language to them and to whomever supervises them. IMHO.
Good luck 