Hey Drew, Cant recall any power points in the shower blocks and nothing mentioned on the QPWS sites ., hand wash clothes only in dishes or buckets, the sinks can get a work out from people washing up greasy plates / kids etc ..
From what I understand the showers it the QPWS camp grounds are now $2 [ coin ] per 3 minutes [ Take a good quantity as change is limited on the island ].., Make sure you turn the taps off before dropping the coin in .
Cathedral beach Resort has powered sites and washing machines, but you would have to be booked into them to use ..
Think you will find a hand wash in cold water and a good wring out will be more than ok for your clothes, They realy only get a bit sweaty is all , and they will dry fairly quick .