Honestly he's doing fine by the sounds of it - as @LandSharksAnonymous says, he is only teeny still. Does he sleep alone at night?
My boy is ten months old, half a spaniel, and would very much like to be attached to me at all times. At five months I was convulsed with anxiety - if you go through the puppy threads from about five months ago you'll see all my posts panicking about the fact that I was panicking 😄
When they're this small lots of dogs really want to be with their people all the time, and the DTAS FB page (Dog Training Advice and something I can't remember!) which is run by trainers recommends that puppies aren't left alone till much later than most people think - not till nine months or so. My pup has never been left fully alone for more than a few minutes even now, he's always been with a sitter or another dog, and I'm still not in a hurry to leave him completely alone. I know others disagree, but I don't think it's something to be massively concerned about yet.
I think maybe separate out all the worries, as five months was for me peak anxiety time. For the groomers: grooming is in itself stressful, so if you can brush him at home then just avoid it for a bit and take the stress away, until you've got a specialist. For the car: try him in the back with the kids and see how he does. He might just want to be with people. For alone time: only leave him for multiples of seconds at a time, and watch on a camera (they're only about £20 off Amazon). Build up really slowly to five minutes. Make sure you come back before he gets upset, i.e. the second you hear him whining or see him stress-yawning.
Sitters are awesome and can give you a bit of your life back, and a good one will make him much more confident about strangers, even if he worries a bit when you leave and tries to bite your nose off when you return. It's just a matter of very gently expanding his horizons in ways that don't frighten him so he realises he's okay. Honestly I think you are doing fine xxx