Mary, welcome to FT.
The comparative info on accident rates can be found
here.
Ok, so that's the figures. When I'm assessing whether I want to fly an airline from a safety point of view, I think of three things - the airline, the equipment used and the regulators. BA has a generally good safety record, as does most of the equipment it uses. You are correct in saying that the B777 has not yet had a fatal accident - OTOH, it is still relatively new in service. Yes B747s have been involved in fatal losses - but one thing to remember is just how many B747 flights have operated around the world. It is a very commonly used plane type, so its not that surprising there have been a very small number of accidents, given the length of time it has been in service. At the same time, because of that, its safety record comes with the proof of long testing. You are correct in saying 4 engines are better than two - OTOH, to get certified to operate on two engines transoceanic, during the certification phase (the process by which engines are tested by regulators to ensure that they are safe to operate), they would have to undergo additional testing before they are certified to cross the Atlantic. And BA are regulated by the CAA, who take their job extremely seriously.
You're also confusing the state of the inside of the plane and the comments about it looking old with how the plane is maintained. If you think, the interior of a car can be shoddy, but it can still work perfectly. Maintenance cycles are regulated (ie laid down as rules by the FAA and for BA the CAA) - replacing cushions are not

At the moment, the B747-400 fleet is still pretty much in the prime of its life. And with an airline like BA (and indeed most major international airlines) maintenance cycles are regulated closely, and if the regulators (Department for Transport, Air Accident Investigation Branch, Civil Aviation Authority) feel that there is any slippage in this regard, they will intervene quickly.
So for me (and if it helps any I worked in flight safety issues for a number of years) I do consider most of the major airlines in the world completely safe to fly. B747s and B777s are some of the safest airplane types in the industry. And the aviation industry is one of the most safety conscious and heavily regulted industries, with respect to safety.
As to the question which is safer the 777 vs the 747. I'd take a 747 everytime. It's more comfortable
I hope you manage to make the flight - doing something of which you are afraid takes enormous courage. And if you do do it, I hope you manage to have a fantastic holiday, which will show that there is a good reward at the end of it!