OP you want a wealthy man so your life can be more comfortable. Which means you will be dependent on him. This means you will find yourself in positions where you'll have to compromise - and he likely won't - because you'll want to stay for the lifestyle. Think about what your integrity and self-worth mean to you. Honestly, I've met women for whom it's less of an issue, but they often come from backgrounds where they didn't have a professional job to start with.
And if you're thinking of meeting this guy at the gym then you'd better have a body that can compete with a 21 year old former gymnast/ballerina who has moved onto CrossFit. Because if he's looking for someone at the gym, that's your competition. That also means her looking for a trophy wife. Not sure that's what you had in mind? You'd also need to provide a child pretty quickly to tie him (his finances) to you long term - and then get your figure back fast.
Of course there are the men who are looking for an equally driven partner abc would be attracted to your passion about teaching. Sorry to say but a fair few of them would like the financial divide between you as it would make them feel superior (they'd not speak about this openly - usually). Again, is that what you're looking for?
I've seen women do what you're wanting and all I can say is, in capitals PROCEED WITH CAUTION.
Remember too that he'll have family and friends who will assume you're after him for his money. How's that feeling when you're out for dinner together or visiting them?
The easiest way for you to become wealthy is to make the money yourself. Yes, it involves a lot of time investment, but marrying a wealthy man as a goal (rather than happening to meet one) has hidden time, energy and financial costs too.
So how could you become wealthy at 44 as a teacher?
- Career change to one you maybe like less but brings in more $$.
- Start up a business on the side - an exclusive tutoring business that employs others targeting wealthy families could be one direction to look in given your background and proximity to wealth.
But actually I'd say that you are already wealthy. You have a job you love that pays enough - although not loads. It is one that is hectic but also gives you time off (unlike the dirt if men you're thinking of who will never, ever be without their phone for 24 hrs).
Rich men are not some kind of prince who solve life's problems with their money. They arrive with their own problems (addicted to work bring one). Their money can actually add to the problems, especially when there's an imbalance.
I'm ignoring drive and ambition because that's an aside - you can have tons of it and not get wealthy. So you're not looking for that, you're looking for the ones who made it big through work.