In addition to the good advice already given, I want to add that a steadily illuminated Check Engine Light (CEL) is really not an emergency situation, and it actually falls into the “check it and fix it at the earliest convenient opportunity” category. On the other hand, if your CEL is flashing/blinking, that IS tantamount to an emergency. A flashing/blinking CEL indicates that a misfire condition is taking place in one or more cylinders, and it must be rectified immediately in order to prevent very expensive damage to the engine and/or the catalytic converter.
But, I also want to respond to the OP’s question, “how safe is it to just double check the gas cap and keep driving?” Not only is doing this safe, it is really what needs to be done as the first order of business. However, it is so easy to properly secure a gas filler cap that I am mystified by how often people seem to run into this type of problem. I would say that, when you consider all of the tasks that one might have to do in a day, properly tightening a gas filler cap is perhaps the easiest one.