Yep... sounds like common swirls... it happens from washing.
To remove them you're going to need a machine, no way around that. If you'd be happy with an improvement, but don't want perfection then you can work by hand.
Also a big thing is to address the source of the problem in the first place. Does no good to spend time/money to fix it if your wash routine is just going to reintroduce swirls again.
You might find these videos helpful and if you have any questions about product or technique I'm happy to help:
Instructional Detailing Videos