Wheel and tire sizing for 2004 F250 4x4 not lifted

Hi there, I'm new to the 4x4 world so bear with me. I have an 04 F250 4x4 extended cab truck that I don't want to lift but want to change out the stock 265/75 R16's to something a little bigger.  I want some meat on the tires instead of the "street" look that many people are going with. Basically I just want it to look a little beefier and "less stock".  I know that the larger the rim the less meat on the tire, so going up to an 18" rim will only allow a thinner tire right? I don't want to put out the money only to have the 'rubber band' tire look on my truck. So should I just stay with the 16" rims and get more of an all-terrain tire or mud tire to get the beffier look I want?  I don't want the tires to rub and am not familiar with 'off-setting' of rims, etc.  I guess I'm just a little clueless right now and trying to figure out what my options are. Thanks.  

I have a 01 f250 with stock rims and run 285/75/r16 tires with no problems you could possably go a little wider but not much taller. hope that helps