Above Oxford the river is quiet, partly due to all the big cruisers not being able to get under Osney bridge. There is plenty of casual mooring on fields, basically as long as there is not a no mooring sign you can stop, but the mooring locations that others have used will be obvious from the bank. Really the only place above Oxford that you will need to pay is at Lechlade, last time I was there it was £5 a night on the field on the left before the bridge.
In theory the locks are manned during the day but in practice lately it has been less than 50% above Oxford that are manned, higher percentage the further down the river you go. You just buy the visitor licence at the first manned lock you come to, probably Eynsham in your case assuming you use Dukes Cut to get onto the Thames. You fill out a form, possibly the hire company will provide you with the info you need to the form, eg, the manufacturer of the boat and year of manufacture, unless there is a exception for the details for hire boats, I don’t know. The locks all take credit cards for payment.