Apologies in advance that I have no idea what a 6 year old Dawes Giro 300 has in terms of components, so I shall assume your "old fashioned" adjective meant the headset is threaded (to confirm see this).
As you can see from Sheldon's article, replacing a threaded headset is not the easiest of DIY bike maintenance projects. Removal of the old and installation of the new can call for special tools (although some folks do use old pipes, hammers etc. to improvise...). Also it is critical that the new one is of the correct stack height for it to work, so you have to be careful of what you buy.
A Campag Record 1" threaded headset, which is super light and beautifully made, retails for a surprisingly modest £40. However much of it is made of light alloy, so I would definitely not recommend installing it without the special tools.
i fit and replace all my headsets. it costs me nothing and i did use a lump of wood and a mallet for many years, to press the cups into the frame.
A shop won't charge you much, but it something anyone could do with some common sense!. IIRC there is a good guide on bikemagic oh now to fit a headset without tools
If you do it yourself, as I did about 2-years ago on a Giro200 (about 2004 vintage), it won't cost a lot. I used an Acor headset from my LBS (about £8 I seem to remember). I cobbled together my own press with a threaded rod and plates from B&Q. The problem I had was finding a headset with a short enough stack height.