Oh OP, I hear you on all counts, and you've had such brilliant advice already, but on the work front, I absolutely get it. It sounds like you desperately need a break to catch your breath and refocus, but it's impossible as the work will keep piling up.
I'm in the same situation - currently chasing my tail endlessly, and often feel paralysed and so I lose the day doing other things which puts me further behind.
My advice (which I'm also going to follow):
Get the cleaning stuff and clean and polish your work station - make it look nicer and more pleasant to be in. Tidy desk, tidy mind and all that...
If you have three projects that have fallen behind, which can you make the most progress on in the least amount of time? Focus on that.
Make a list of everything that has to be done - write everything down and then prioritise in order of urgency and some quick wins. What would make you feel most happy to get ticked off the list? Focus on those, and just work your way through the list, knowing you can't get everything done in the blink of an eye.
Work this weekend - use some of Sat and Sun while clients aren't working to catch up so you feel more in control of it and don't feel sick just thinking about Monday.
If you're really worried - can you drop your clients a line, say that you've had to sort out some personal issues which has meant you've worked a little less than usual, but it's now back to normal and they can expect the next phase of the project on X day. If they're really understanding, buy extra time and then deliver early - they'll remember that you overdelivered.
When it comes to actually doing the work, make a hot drink, light a scented candle, put on music or a podcast or whatever works best for you, and section off the time and dedicate to tasks, then treats. So, Sat morning, 9.30-10 - write out tasks and prioritise, 10.30-12.30, work on task 1, 12.30-1ish, have a lovely lunch, 1ish-3ish, work on task 2, 3ish - have a nice coffee, and so on. Blitz through the tricky stuff and get things ticked off the list, it will get your brain back into productive mode and you'll feel 10000 times better!