Not sure about others but I'm shitty with his I'm too cool for you attitude and his lack of effort he makes with Tully.
I think it's pretty hard to say whether he's made an effort or not. Drew isn't a big SM user and prefers to go about his life away from it mostly, unlike Tully who feels the need to share every second of her life. I can see how that might give the impression that Tully's made more of an effort simply because she puts everything out there...but we have no idea what they've talked about privately or who's initiated it.
Also like @smurfette already pointed out, there's no reason to think Tully has gone to Melbourne for Drew. Actually due to the fact her first night there she was out with Ed and Nick instead of Drew, and also that Drew might be leaving again indicates she's not there for him. She doesn't seem to have much going on work wise in Sydney so she might be down there because she has something going on or wants to spend time there looking around for some potential work.