Maybe my experience will be useful to you.
I am in Boston, and I have had my own roofing business for 2 years, and my own carpentry business for 3 years. I've also worked for other carpenters for about 4 years combined.
I applied to the union over a year ago. Here they have an 'apprenticeship and training program' -- maybe this is national.
They admitted me and invited me to apply to a local. But they admitted me at apprentice level. The pay would be $17.50/hour plus benefits. Actually that seems like a pretty good wage, even for Boston where cost of living is the highest in the nation. But I didn't want to become an apprentice after I already paid my dues for around 8 or 9 years.
I probably would have moved up to Journey fairly soon after starting, but I just chose to stay in business for myself and also work with another non-union company when I didn't want to handle the business side of things, sometimes.
I'm sorry you're having trouble getting paid - maybe you could also check into some other companies that have their business more in order.
It's just plain stupid for a company to delay paying their hardworking employees. All it requires is good organization, a good payroll service, and enough operating capital to pay everyone each week.
If a company don't have any of these things, then it's badly run or it's going bankrupt.
Anyway, if you do go union, have all your experience documented, like a resume, so that they see in black and white all your experience, and don't accept being admitted at apprentice level if you have 10 years experience. Be firm and confident. If the market needs union carpenters in your area, they will want to admit you.