Follow what DaveP suggested it will work for you.
While you have the needle plate out get a lighter that is lying out and burn the wick until it glows bright red and when you remove the flame it should be white. It is made from a high flame resistant material it will not melt.
Once that is done there is a video on Youtube that will show you the proper adjustment of the wick.
Yes, I, too, recommend the atty washing procedure shown in the above video. Made mine taste and draw like it was new.
Please for the love of everything holy DO NOT take the wick out of the needle plate it is HELL to get back in there, I have destroyed a wick doing that.
If you must there are two threads on here on replacement wicks:
Method one is using a zippo wick, I have not tried it myself but other people rave about it.
Method two is getting some silica rope and making your own wick with it.
Be careful when dry burning an eGo-T atomizer they are sensitive and will break.
Most times you will not avoid pulling the wick out of the needle tube because it's stuck to the coil. To get it off the coil use a fine pair of tweezers and gently pull it off the coil by gripping the wick material below the metal cap.
DO NOT PULL IT OFF BY THE CAP or you'll pull the wick out of the cap.
Reinstalling the wick into the cap is very easy, too. Just push it in from the bottom, cap end first, and then adjust the height so the top of the wick is just flush with the bottom of the slant of the needle.
Below is a video on putting the wick back into the needle and adjusting it: