Yes, until mid-century, anywway. That much I know. Cross belts distinguished regiments of Horse until at least 1746 when they became Dragoon Guards, after which I believe they did their best to avoid being classed as Dragoons in everything but name. DId that include retaing their cross-belts?
in 1775, the 8th Dragoons, inter alia, were converted to Light Dragoons, who were all to wear their swords from shoulder belts. This deprived the 8th Dragoons of the unique 'privilege'(among dragoons) of wearing their swords slung in that manner (a distinction supposedly dating from the WOSS, ; tradition shaky). The nickname 'St George's Cross-Belts' was still current at that time. This distinction, in the form of buff cross-belts was restored by order of King George in 1795, in recognition of the regiment's exploits in Flanders under the Duke of York.
The not-entirely-reliable Richard James wrote in his 'New and Enlarged Military Dictionary' (1802):
"BELTS… the horse to have cross-belts; the dragoon-guards and dragoons to have only one shoulder belt, except the 8th regiment, which is permitted to wear cross belts."
It reads as if Jsmes was quoting from an official order, 'the Horse' perhaps referring to the Royal Regiment of Horse Guards (The Blues), otherwise I am not sure how there could be reference to "The Horse" fifty years after the conversion to Dragoon Guards.