I am having reeeeeeal problems continuation betting against very loose players like people with a VPIP of 40%+. I raise with legitimate hands on the button if it's folded to me like A high, K8+ but obviously 2/3 of the time I miss completely. The problem is if I CB I mostly get called because these players literally call a PSB with any pair, flush draw, straight draw (OE or GS) and sometimes even overcards. This makes taking the pot down on the flop very difficult. If I check my AT behind them on a J54 flush draw board for example I often face a quick PSB which I have to fold unless I hit my A. The same happens if I've raised out of position with AQ for example and I check to them. As a result I find myself wanting to dump my A4 even when it's folded to me because I know I'm going to raise and probably will have to fold it on the flop.

Does anyone have any advice? I am sure that I am a favourite against their range but the problem is that they can't be bluffed and for that reason I think it is dangerous to play back at them when I suspect that they have nothing. Thanks