There really is a lot of good advice in this thread. I, myself did things in a very unconventional way, and jumped in feet first with no experience. Here's a link to my story about "jumping in with boat feet". From Newbies to Seasoned Cruisers
I spent 10K on a 31ft sinking wooden boat. It almost sank in a near-shore storm, but it didn't and I lived on her for 6 great years and she taught me a lot.
For many of those 6 years I thought she could be an ocean crossing boat. Unfortunately I was so green that, I had no idea. She could NOT be a blue water boat. I'll draw a parallel on the point of experience. I once had a doc tell me, when I was newly out of PA school that I had to be very careful. It was not because of what I didn't know. If I knew that I didn't know the answer, I could ask or look it up. It wasn't the info I knew that was dangerous either. It was the stuff that "I didn't know that I didn't know" that would be dangerous.
In Bjones post he points to only a few reasons why a catalina isn't an ocean going boat. Who would have though that a big cockpit opening is a fatal flaw? It can be. It takes a while and some experience to know these things, which is why it's good that you're asking those who do know.
Even though I'm an iconoclast myself and I jumped in with both feet I don't think you can get a good blue water boat, that you
will be happy with (remember you have to figure out what YOU want in a boat i.e. big and slow vs fast and light) for your stated budget. I would LOVE for you to prove me wrong and I'd happily eat my hat, but I fear it's not realistic at your current budget.
I see 3 competing things that can't be reconciled:
2. Blue water capability.
3. Less than 10K price.
I would argue that if you only choose 2 of the three above, you will have more choices than you can shake a stick at. But it'll have to be 2 of the 3 above.
Lets see if we can look at this a different way: How badly do you want to actually cross an ocean on your own bottom? Is it the crossing itself that attracts you, or the destination (Europe?). You could sail your whole life up and down the US east coast and into the Caribbean and never anchor in the same place twice. What about skipping the Atlantic crossing? If you skip the crossing requirement you could get a great coastal boat and enjoy costal cruising for as many years as you like. You don't have to cross an ocean for it to be a cruising life.
If crossing the Atlantic under sail is really your dream, what about getting passage on another cruising boat that needs a third person as crew or on a race boat?
If sailing your boat in Europe is the dream, you could buy a coastal boat and have it shipped over one day, negating the need to spend all the money on a sturdy blue water boat and all the gear. Shipping your boat over to any spot on the globe is not as expensive as you think.
If it absolutely has to be one boat only, and it has to be big, and you have to cross the Atlantic on YOUR boat, then price has to go. That can be done though. You could get a boat loan. I have a Formosa 41 and they're HUGE on the inside and are awesomely seaworthy and good ones can be had for as little as 30K. It's three times your budget but you could do it with a boat loan.
Also if you don't want to go into debt, and you want to cross on your boat, it'll have to be size that goes. Go small and go now has worked for many.
Pick 2 of the three and the dream is doable in one step as you require.