What happens when you need to play one hand only? Or when you need to play with hands independant? As you said, HS and HT feels different. Also, don't you find it hard to concentrate on getting your technique right with so many things going on at once (with both hands doing their own thing)?
I think you are right.
However, the original question was “How do you practise?”, not “How should I practise?”.
So Lostinidlewonder described his way, which happens to go straight to hands together. Please note that he is a concert pianist with many years experience, so
he can get away with it. A beginner struggling with basic motions will not get away with it. If you have not figured out the “technique” hands separate first is a must.
At the same time “practice” is not a rigid procedure. It depends on the passage, it depends on the player, it depends on your mood that day, and maybe even on the weather outside Pretty much what Steinwayguy said in reply #3. Therefore you do not do hands separate as a religious observance, but on an as-needed basis. For a total beginner, the whole piece may have to be learned with separate hands before he joins them. For an intermediate student, many passages of a piece will be tackled with hands together straightaway and just a few will need to be addressed with separate hands. An advanced, accomplished student (of pianist) may be able to forego hands separate altogether.
A huge mistake many beginners make is to try to emulate the practising ways of the great pianists
as they currently practise, rather than enquiring how they started and eventually ended up practising the way they do now.
If I tell you that accomplished readers do not join letters into syllables and syllables into words, but rather “recognise” full words by their pattern, this does not mean that an illiterate person wanting to learn to read can skip the phase where s/he recognises (and memorises) the letters, learns to join them into syllables and so on. It is part of the process, a ladder so to speak, that will get you to the next level, and that can be discarded only after you reached the next level (I have been reading too much Wittgenstein). So hands separate is a very necessary ladder, but by all means discard it the moment you do not need it anymore. However, if you do need it, then going straight to hands together is going to be a major disaster. You can read more about this here:
https://pianoforum.net/smf/index.php/topic,3085.msg44855.html#msg44855(Hands together – dropping notes – when to learn HT and when to learn HS)
https://pianoforum.net/smf/index.php/topic,4123.msg37829.html#msg37829(several ideas are re-explained and reinforced in this thread – How to investigate the best movement pattern: Example Scarlatti sonata K70 – How to work out the best fingering. Example: CPE Bach Allegro in A – Slow x slow motion practice – HS x HT – practising for only 5 – 10 minutes)
https://pianoforum.net/smf/index.php/topic,3085.msg27140.html#msg27140(Hands together: when and how – dropping notes)
https://pianoforum.net/smf/index.php/topic,2802.msg24467.html#msg24467(When to join hands – Bernhard explains it all)
Now to answer the original question:
1.longs runs really fast – Combine repeated note-groups with chord attack.
2. big chords and not stopping to look for every one. Start by looking and use a rotation movement of the forearm combined with hand shift guided by the arm. Ignore the notes completely (it does not matter if you make mistakes), and instead
practise the movement. Now fine tune the movement so that it becomes completely accurate and you are always hitting the right notes. Rotation is the basis for accuracy, and as such is even more important than the hand shift. Difficult to describe, easy to show. The movement should be circular (or elliptical) on the horizontal plane, and arch-like on the vertical plane, so that when combined it is like a diagonal circle. As you can see, the notes and reaching for them – which is what most people practise – is not really the focus here. Once you get the hang of the movement by looking, start putting your attention and awareness in how it feels internally, that is the physical sensations needed to always land you in the right place. Then stop looking (or use peripheral vision) and use your ears to get you to the right place. Finally do it with eyes closed.
3. sight-reading – Accompany people. Play duets (even if you are a terrible sight reader, you can do it if you choose a repertory easy enough, e.g. John Pitt’s recorder books for children which has very simplistic piano accompaniments). By the way, this is what I do, not necessarily the way for anyone else to do it.
4. just reading music in general – Just read music in general. I have a huge stack of sheet music of different levels, and I am always reading through it in search of hidden gems (especially if they are easy and sound difficult

).
Best wishes,
Bernhard.