If you are getting along well with the parent who is housing you, there is absolutely nothing wrong with continuing that arrangement for a while longer. That you are paying rent to your parent makes this arrangement beneficial for you both. I think you'ld be very foolish to move out right at the moment.

Your sensitivity and depression are a separate issue. I think staying the course, as you've set things up, makes sense. Never base your plans on what others think you should do. When people pass comments like you described their motivation is not pure. They are being catty and deserve to be thought less of.