Ronan has it right, from a trawl through my bookshelf.
Dutton's were at Edgehill, and their Colonel was Sir Ralph Dutton, Lt Col Stephen Hawkins. Hawkins took over the regiment around the turn of 1644, probably early spring of that year. The regiment were involved in the June/July in the defence of Greenland House near Hambleden in Buckinghamshire, part of the actions of the First siege of Oxford in that year. According to one source, it seems as though the reference to white coats dates from the September of 1644, and it's often apparently misquoted as being the case as far back as Edgehill.
As Ronan suggests, at Edgehill, many of the Royalist troops might have been in civilian kit. That said, the Duttons were apparently an extremely wealthy family from Gloucestershire, and his regiment was about 800 strong when he joined Charles at Nottingham. There's nothing wrong with having troops in white to start off with, therefore - simple woolen uniforms from a wealthy colonel seems perfectly ok in the absence of any hard evidence!
As for later, it seems as though the regiment was at Bristol, Newbury 1, and likely the Relief of Banbury in 1644. Work on the new coats for the Oxford Army began in the January of 1643, and Duttons may have been in the main Royalist camp at Abingdon at that stage, so might have received some of the red or blue uniforms, before a later refit of white.
As for colours........ any body's guess! Yours look fine - stripey colours were used, a kick over from Tudor era stripey colours, popular with local and militia forces in later years right up to the time of the Civil wars.
So to sum up....... Edgehill, most likely civvie clothes, but nothing wrong with white. 1643 early 1644, likely red or blue, late summer 1644, highly likely in white. Colours, yoyr choice!
Hope this helps!