The 20-year-old agreed terms with Ton boss Moore just under two weeks ago and finally signed on the dotted line at Cappielow yesterday.
Fellow defenders Mark McLaughlin and Scott Taggart also inked their new deals at the same time - and O'Ware is aiming to play alongside them more regularly in the upcoming campaign.
O'Ware, who made just 10 starts and five sub appearances last term, told the Tele: "I signed a new one-year contract down at Cappielow yesterday.
"There wasn't really anything behind me holding off from agreeing a new deal at first, I just wanted to take my time, think over it, and make sure it was the right decision.
"I'm looking forward to next year now. There are three or four weeks of summer left, so I'll enjoy them and then, hopefully, I can make a claim for a start right from the beginning of pre-season. That's my aim. I'm settled in after two seasons here and this year I want to be playing in every game.
"In my first year I didn't progress that much, maybe just a wee bit, but I think I kicked on last season playing alongside guys like Mark McLaughlin and Craig Reid, who were both very influential.
"So hopefully I can build on the progress I made last year. This is my season. I want to be in the team starting every week, and if I'm not in the team I want to be pushing hard to be in it.
"I see that being at centre-half. Obviously the gaffer knows I can cover a few positions, but, first and foremost, I'm a centre-half and that's where I want to be playing."
Meanwhile, boss Moore has revealed that there was no breakthrough despite his discussion with the agent of out-of-contract defenders Reid and Willie Dyer yesterday.
The Ton boss said: "We don't seem to be any further forward with regard to Craig and Willie in all honesty.
"The agent says he will now speak with them to see what the situation is.
"Willie's away on holiday at the moment - he's away to Egypt, I think - so we don't know what the situation is. We'll just have to wait and see."