Lawson made only his fourth Scottish Premiership start for Motherwell in a 2-0 win at Rugby Park, although he has also started four games in cup competitions.
The 29-year-old is likely to keep his place for Saturday's William Hill Scottish Cup fourth-round clash against Albion Rovers after putting in his best display in a Motherwell jersey.
"The standards haven't been set high but it was good to get a full 90 last week," he said. "I thought I did all right and it was a good performance from the boys generally.
"Competition is high but I set my own standards and I've not been happy with my performances so far.
"I think that has shown because I have been in and out the team. But hopefully now I can get a good run in the side."
Lawson rebuilt his career in Dingwall after being released by Celtic and proved himself in the top flight last season after being instrumental in County's surge into the top six after Christmas.
And he is comfortable being in the position of having to win over the Motherwell fans.
"It's something you have got to do," he said. "I knew when I came here that I would have to prove myself all over again. I have got to keep working hard and prove to the fans and everyone that I'm worthy of being here."
Lawson's honest comments mirrored recent remarks by Iain Vigurs, who followed his former County team-mate to Fir Park. Vigurs also impressed at Rugby Park although a thigh injury may rule him out of the Lanarkshire derby, which has been switched to Hamilton's New Douglas Park.
And manager Stuart McCall knows Lawson and other players are working hard to hit peak form.
McCall said: "A few lads started the season well and had a little dip, a few lads have maybe not got going to the extent they would like to do.
"But we all know what each other is capable of. There is a good team spirit among them and everyone is wanting each other to do well as well.
"You have to take your opportunities and I thought Lawson was excellent last week and showed what a good passing range he has got.
"Ross County played with three in the middle of the park and we don't always do that so it's slightly different for him. But he is a brilliant professional and he is working hard every day to force himself in the team."
Motherwell have never lost in 34 competitive meetings against their Coatbridge neighbours outside the Lanarkshire Cup and McCall has urged his players to draw on recent experience of playing less-fancied teams.
Motherwell have put six goals past Morton and four past the likes of Dundee, Queen's Park and Clyde under McCall.
He said: "It's about your attitude and how you approach it. Because as much as though we should be technically better on the ball and be able to move it quicker, they will certainly be able to match us for aggression and work-rate and energy and that's something we have got to match.
"It's about your application and starting on the front foot."
Albion will be without on-loan striker Mark Shankland after parent club Ayr turned down a request to field him.
Fellow loan signing, Dunfermline's Scott Mercer, has been cleared to play and could be involved from the kick-off as Peter Innes (leg), Kevin Nicoll (knee) and Pat Walker (ankle) all remain sidelined.
Albion boss James Ward said: "Motherwell have been a consistently good side under Stuart McCall and Kenny Black, so this is a really hard tie for us.
"But I'm sure that the players will go out and give it everything they have."
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- Albion Rovers
- Stuart McCall