I think your Dawes will be up for the task, but with a few modifications it will make your job a bit easier. As suggested, I'd dump the rack and get a set of racier, lighter tyres in the 28mm to 35mm range (Schwalbe,Continental etc). I’d also experiment with the height of the adjustable stem – it maybe that you can lower it a little for a more aerodynamic position. I’d also get the bike serviced a couple of weeks before, and then take it out for a few shakedown rides to check all is OK.
That said, given that you are powering the bike, the fitter you are the faster you will go and for pretty much everybody that will be the biggest difference in performance over a 3 day ride.
When I did the London – Paris there were people who were going quickly and there were people going considerably more slowly. It was fine. Some people would stop at bars in every French village and wait to be swept up by the support vehicles – I thought that was a bit cheeky…(it wasn’t me, I promise).
If a new bike appeals to you, I’d consider a touring or audax style bike with drop handlebars, but the important thing is trying different bikes first to see what is comfortable and fits!
Hope you have a great ride.