Absolutely - do tell!
Shipping it makes sense because those carriers will allow you to properly insure it. I think you can have success traveling with it but you are certainly subject to the possibility of running into the wrong employee. I would absolutely not check it with the airline as they will not take responsibility for its safe transport.
No they won't. At least UPS and FedEx won't in the general sense of how people think of insurance. Not sure about USPS.
UPS and FedEx will let you pay extra to increase their limit of liability if they are at fault for damage, loss, etc. The potential problem in this case is that the OP said the document has "sentimental value". As you cannot really quantify that and as it sounds like the document cannot be replaced, UPS and FedEx are going to limit how much liability they will accept to a nominal amount. Whether or not that's enough would be up to the OP to determine.
I do agree with the posters suggestion proper packaging and shipping is the way to go. Or maybe a combination. Buy the artwork packaging so that it is protected and then see if it meets the dimensions to be checked. At least checking it in vs. shipping is going to potentially involve less automated equipment moving the shipment and less people handling it so hopefully the risk of damage would be less.