In theory if you have more than 3 sheep or a rare breed you have to register. But a lot of people dont. However, I did last year and mine went off to the Board.
For six you could choose either to shear them with electric shears you buy, or hand shear them yourself, or try to find a shearer to do it (which will cost the same or a bit more per sheep than you will be paid by the wool board but at least that helps offset it).
Personally I would shear them yourself with hand shears. Im only an average sized female and I did 16 ewes and a ram with hand shears. Dont worry you will cut them, so long as you are careful (and ideally someone shows you basic stuff) you will probably cut them a lot less and a lot less deeply than professional shearers as you can go as slow as you need.
Also you can do it at the right time, early enough to prevent fly strike but late enough that the fleece has started to 'rise', this rise gives an oily yellow line in the fleece a short way up from the skin, and cutting down to the rise line would be enough from a welfare perspective if you dont need to sell the fleeces (it will also leave them with a little fuzz which can be useful if you shear and then it rains for 3 weeks or is very cold).
My Shetlands dont really shed their fleece apart from odd bits so they are sheared, it depends on the genetics, also shedding creates a hell of a mess so I prefer to shear.
NB Horse clippers rarely have the power to do many sheep without dying (plus it can be hard to get the sheep combs to fit), I bought sheep clippers even tho I have horse ones altho I still normally either hand shear or get a professional shearer in depending on the numbers.
If you let us know rough area/region we might be able to recommend a shearer or teacher (if you arent in singing shearer realm).
Keen-ness is a good thing :-DDD