So ideally we need to break this down into two questions:

1. Were the English or Welsh using castles prior to 1066?

2. Was the use of stone fortifications exclusive to post-Conquest England

So:

1. In short, no. It's thought that there were a handful of castles in England before 1066, although these were constructed by Norman knights who had been invited to England by Edward the Confessor to enforce the Welsh Marches. Prior to the Conquest, English fortifications were still based largely on the burh system which had been so crucial to the Danish wars of the 9th and 10th centuries. Castles and burhs, very broadly speaking, both serve the same primary function: the control of territory by mobile garrison forces. The differences in their relative designs comes down primarily to differences in English/Welsh and Norman warfare, but also in part to the relative social positions of English and Norman landholders.

English warfare pre-1066 is predominantly infantry-based, centred around the use of a shield wall. The mobilisation of a fyrd garrison therefore required a significant amount of manpower, and the commensurate space at burh fortifications to house them. While the Anglo-Saxons are known to have ridden horses to battle, they don't appear to have made tactical use of cavalry. By comparison, Norman warfare was much more heavily centred around the horse. A mounted Norman force was intrinsically more mobile, and as such a smaller force was able to control a much wider area. A lower demand for manpower required a smaller fortification, which also allowed castles to be built in more physically defendable locations.

The Anglo-Saxons were fond of fortified manor houses, but in the event of a war or a raid, it could be presumed that a local lord might already be at a burh or with his forces. The Norman position in England remained quite tenuous well into the 1080s and 90s, and meant that castles needed to function much more as a potential defence against a broadly hostile local population than an English manor ever did. A stereotypical motte-and-bailey castle therefore allowed a very small garrison to defend against a much larger force if necessary, although like the Anglo-Saxons, the preference was still to engage the enemy in the field.

2. In the immediate aftermath of the 1066 Conquest, Norman castles in England are built mostly of wood. Castles built from stone start to appear largely at the end of the 11th century before becoming commonplace in the 12th. Anglo-Saxon fortifications are largely built of wood, often in the form of a series of palisades and ditches surrounding a settlement or site. This doesn't imply that they were in any way 'lesser': wooden construction allowed for burhs and other smaller fortifications to be constructed relatively quickly by the armies of the time without the need for specialised engineers and equipment.

With that said, the Anglo-Saxons absolutely did build in stone, and made use of stone fortifications, albeit largely pre-existing Roman ones. Exeter, London, Canterbury, Rochester, Rocester and Chester are all examples of Roman settlements where the stone walls were reused during the Anglo-Saxon period. It's worth noting, however, that in many cases, the continued use of the Roman walls required significant reconstruction rather than just simple upkeep. Prior to its reoccupation in 907, for example, Chester was noted to be entirely in ruins, and the Mercian restoration of the city was feted at the time as a major triumph of Mercian logistical and technological capability.