We did 12 month back packing in our early 60s after retiring. Part of this was in Australia and New Zealand.
I didn't find washing a problem, I just used the facilities available locally.
We camped in tents, stayed in hostels, hotels had a motorhome for a period. We camped in hot weather just put a line up.
If you are booking hostels again it's not a problem, ask at hotels before booking what their facilities are.
Always consider the sort of clothes you need and ones that don't need ironing and the weight.
Take advice on Malaria tablets if you might need them from your GP or any other medication you might need jabs for. If you need to take large amounts of medication ask for a letter to cover this.
If you are traveling the west coast don't forget to carry extra petrol and water the garages are miles apart.
Have a great time.
Nicola Sturgeon’s husband re-arrested