Hi SHIVA, good to hear from you again!
Yes, all DSLRs offer Live View and several have articulated screens, but as you fear, their AF in Live View is still much slower than most mirrorless cameras. So if you're used to the GH1, I'd stick with mirrorless.
The GH3 is the natural upgrade for you, but if pro video isn't your priority, consider the Olympus OMD EM5. It uses the same lenses as your GH1, has a vertically tilting screen, built in stabilisation which works with any lens you attach and is weathersealed.
The Sony NEX 5R, 6 and 7 are also very nice, especially now there's an ultra wide lens available. They also have tilting screens, although again, just vertically.
If you want a screen which flips and twists on a mirrorless camera though, it's really down to Panasonic, with the GH3 and G5, the latter being a great choice if you don't need pro video and weather-sealing.
Should have a NEX 6 and G5 review soon.
Here's our Olympus EM5 review