You could try Ward Bros (Malton) in South Bank. Ancient fleet (they gave me a 54-plate DAF when I did a Palletways shift for them a week ago), but well maintained. I’ve just applied with them this morning for a C+E job. They are often desperate for drivers, especially C+E! If you visit their yard with your CV, the Palletways office is on the first floor via an external staircase.
I had the same trouble as you when I passed my class 2 last year. Took me about half a year to get work.
You’ll get there sooner or later.
Good luck! 