Narrow-wheeled Bicycle, for getting to work (Wolverton MK12)

I've used a narrow-wheeled (used to be called a Sports Bike - with drop handlebars) bike since I was a teenager ... now in my 60's. The current bike is past its sell-by date and various bits are thoroughly worn out - most notably the bearings, the chain and block system, and the wheels and tyres (28"x1.3/8").
Alternatively a bike of this sort for spares-or-repair might make it possible to make one working bike out of two.
The current one makes very 'I'm almost dead' noises and the wheels wobble something terrible and the chain falls off periodically [all despite intelligent care and lubrication].
Many thanks for reading.