I would caution you that the above is not necessarily "simple." There is quite a bit of strength and reach required to move the lever through the full range of 7 gears, and of course you may have to readjust if you did not get it just right or vibration moves the lever. Also, if there is a problem with cable binding you will have exactly the same issue as you do now.

Actually I did not address one issue I should have. If you purchased this last fall, was it bought new from a bike shop? Generally there is a free adjustment period for all bikes, and even after that a minor problem like this may be addressed at no charge.