Sometimes it takes a change of "strategy".
Try this: Go ahead and start right off with both thumbs. Type deliberately slow at first, focusing entirely on the keyboard itself and where your thumbs are landing, and not so much on the words above. When I say "type slow", I mean to pause for a half-second before lifting your thumbs after each letter. Remember, you don't have to tap hard -- in fact you can tap the screen very lightly. Just slow yourself down until you get a rhythm.
Then, start picking up the pace. Learn to look up at the words only occasionally, such as when you finish longer words but before you hit SPACE, to make sure it came out correctly and to cancel the Autocorrect if needed (especially after odd names and words that might not be in the dictionary). But focus mostly on the keyboard. It's counter-intuitive for us touch-typers, who are used to looking only at the screen on the PC. But you'll get the hang of it.
Also, learn to trust the autocorrect. If you know you've made a mistake but the word is mostly correct (and it's a common word), just keep going, and glance up to verify it came out right. I always give my sentences a double check read before I hit send too.
Finally, try landscape orientation. You might like that better.
Those are my tips.