The part of the sentence "obsessed with doing" is correct and usable in written English.
You can use it to describe someone who is preoccupied and very focused on doing something. For example: "She was so obsessed with doing her job to a high standard that she worked late every day."
In our exercise-obsessed nation -- either obsessed with doing too much exercise or with our guilt about doing none at all -- the role of exercise in promoting sleep has been entirely absent.
"I got very obsessed with doing laundry," she said.
"It was a huge decision, but I was obsessed with doing this".
During a sequence filmed in Italy, Hammond was obsessed with doing "doughnuts", dirtily circling a car in a Florence square and later doing ovals in a speedboat in Venice lagoon while Clarkson and May look on, horrified, from a gondola.
A great many of us view him that way, and more — as a man not simply "willing" to use force but obsessed with doing so, who makes Richard M. Nixon's "imperial presidency" seem a model of constitutional governance and George W. Bush's deceptions by misstatement and nonstatement seem like reckless candor.
A sour, rasping-voiced old lady who hates life so much that she tries to ruin everyone else's pleasure; an anxious teen-age girl whose Sarah Bernhardt emotions center on zits and jeans; two women so obsessed with doing the right thing that they become anhedonic wrecks.
A team with five debutants was so obsessed with not doing anything wrong, they forgot to do much right.
You see, we, like many Americans, are intent -- no, obsessed -- with doing the right thing when we travel abroad.
We're obsessed with this "Do The Quad Solve" rap video -- they gonna show you how to do the quad solve, all you gotta do is watch 'em solve!
"The questions that I'm obsessed with just don't bother other people as much they bother me," he said, in his office.
He conceded that a country obsessed with winning does not always see the silver lining -- especially when there is no bronze lining, either.
