I had a KLR650 for around 8 years, a great bike that did everything I wanted of it, certainly a better bike than I was a rider.
I spent a bit on mine by beefing up the suspension (newer models are a big improvement over old I'm told), crash bars and making some hard luggage for it (not recommended, it is secure, but hurts when it hits you).
Dunlop 606's, were my tyre of choice, I tried a few others but always came back to the 606. Not the best on road, but gave me confidence on the dirt and wore reasonably well. Down side is they are a bit noisy. As for expensive, I was getting them for around $160 (Topgun Yamaha in central Ipswich) a set and fitting them myself, so hardly breaking the bank.
So, DR v KLR?
Both do the job, the DR is a bit more dirt orientated and air cooled. KLR is water cooled, so a crushed radiator will leave you stuck and a set of crash bars is around the same price as a radiator.
KLR stator puts out more power, so can cope with heated grips, lights, GPS and phone charging easier if you are into your electronics.
Neither is a rocket ship, but the DR might be a bit better as its carrying a bit less weight, but the KLR has a better fairing.
KLR has a bigger tank, but a 5L fuel can adds flexibility and costs bugger all. Safari also do a massive tank for both if the budget extends so far (on the KLR, with the big tank protects the radiator as a bonus and eliminates the need for crash bars, which is good as they won't fit anyway).
Some KLR's used oil from new at high RPM when cruising, ballancer shaft tensioner spring losing tension/ failure (the dohickey) DR's had output shaft failures.
Both have a a lot of after market stuff available to make them into the adventure tourer you want and will do the job easily. There are shedloads out there doing it everyday.
Toss a coin, spend some coin soft luggage, sorting the suspension to better suit your weight and the load and then hit the road.
And since you are in Qld; if you don't carry a pilion and don't intend to, chuck the rear foot pegs in the shed and look at getting single seat rego on the bike.
Single seat:$315
Two seat: $485
