North Railway Station to Chengdu Shuangliu Airport

Arrival by Subway line 1:
Passengers arriving by subway should get out from exit A1 to the station.

Arrival by taxi
If arriving by taxi, passengers will be dropped off at the front square.
From/ To Duration (minute) Cost (CNY)
East Railway Station 25 30
South Station 30 45
Shuangliu International Airport 45 75

No, you don’t need to the railway stations. You are suggested to take a taxi to Taipingyuan Subway Station of line 10. It takes about 10 minutes and CNY20. Then you can take Subway Line 10 directly to the airport, which takes about 20 minutes and CNY2.

Shall I take a cab to Wenshu Monastary Station LINE 1 to South Station then change lane to Taipingyuan LINE 7 then take LANE 10 to Shuangliu Airport T1.
Or from South Station I can go straight to the airport T1

Yes, both routes are okay.
But I think you should take a taxi to East Railway Station first, since there is bullet train to the airport. And the train takes about 20 minutes.
By the way, after getting to North Railway Station, you can take Shuttle Bus line 2 directly to T1.