"It looks like wrestling a hedgehog" - probably my favourite description of knitting with double-pointed needles from someone nervous of trying it.
There is a misconception that you have to hold all your DPNs all the time. In fact, once you get into your rhythm the needles will stay put unless your accidently pull out the wrong one.
When I am teaching in person I wave my sock knitting round my head to demonstrate how secure the needles are - try it! If you were at the Knitting and Stitching Show last week, you may even have seen me do this.
This means there is no wrestling involved. Just concentrate on knitting across the stitches on the next needle, and only holding the two needles involved in that. Then when you free up a DPN, rotate your work and start across the next needle, ignoring the others. That way you are only ever holding on to and worrying about your active needles. The others will just sit in your work. If you look at the picture, I'm only holding two needles.
The only way to get over that hedgehog wrestling feeling is to practice with your DPNs, after a few rounds it will feel much easier.