I'd wait til next year. i have sown row upon row fortnight after fortnight this year and nowt happened. It was the strange weather conditions. So I have put it down to an expensive waste of time this year.

I grow mine in cell trays before planting out. Never any problems.
If that doesn't work for you, grow something else. Sometimes you have to accept that certain soils won't grow certain crops.

Once germinated beetroot should be easy to grow, so either adopt Scrumpgral's idea of growing in cell trays (though you need to water carefully and plant out before the tap roots are affected) or use some of the 'square foot' gardening techniques.
This involves improving the soil considerably in a fixed area with the addition of a couple of handfuls of vermiculite or perlite (or small gravel or sharp sand would do), a few handfuls of compost, a little blood, fish and bone, and a scattering of lime. Then mark your square foot and make around nine regularly spaced holes with your finger, put a small amount of vermiculite into each hole, sow a single beetroot seed, cover with more vermiculite. Then keep well watered until germination occurs. We get excellent beetroot by this method even though we are on heavy clay soil. Vermiculite or perlite retains moisture so tends to stop germinating seeds from drying out.

I have a list of things to go in cells to start them off next year. Beetroot is one of the large list. I did the paper towel thingy with parsnips this year, only forgot to check for a couple of days and the little roots went a bit curly, but I transplanted them into loo rolls and eventually into the bath in the plottie. They are coming, even if a bit slow. But I pulled one up the other day to check and see and it looked like i had used curlers to grow them. :-)

thanks for your help