You have poor air movement through your turf which is encouraging the moss.
I wouldn't start again because you will effectively ruin any soil structure you may already have.
When the ground is dry, push a garden fork deep into the surface to 'open it up' a little. This should be done as many times as is possible over the entire lawn area.
Then, give the moss a good raking over to remove as much thatch as possible exposing some soil, overseed with a good quality dwarf ryegrass based seeds mixture, roll well, then finally feed at least 2-3 times a year.
Do not use moss killer on new grass seedlings, it will burn them up.
Given that you have very sandy soil which is prone to fertility losses I would mulch all your clippings back onto the ground to recycle as many nutrients as possible. This will also encourage worm activity and keep your soil 'open' to some extent.