A lot of the old commercial shortwave bands are pretty much dead nowadays so your old radio may not be actually faulty....just nothing there to receive
Decent communications receivers were and still are relatively expensive, even older second hand units command quite a price. (Or at least above the budget you mention, "expensive" is a relative term).
The biggest bang for buck receivers are SDR, (software defined radio), usually available as a USB 'dongle' and software, so long as you don't mind being tied to a computer they have a lot to offer. They're cheap, cover a wide range of frequencies, (LW to UHF), and being software controlled have a lot of demodulation options, scanning, frequency input, waterfall display, etc.
I have seen people run these type of units from laptops, tablets and even android phones so a lot more portable than first thought. At sub £20 prices and upwards they're highly affordable for what they can achieve.
For a truly portable, battery operated radio experience most modern receivers are based around DSP, (digital signal processing), technology. For the budget you mention something like the Tecsun PL-600 could be a good buy. This radio has been around for a few years, (more modern and more expensive models are now available), but has a good range of options for short wave listening.
There's too many options to give you firm recommendations of particular radios, it depends on what 'floats your boat'
For me personally, SDR and DSP technology seem to suffer a lot with noise from other electronic/electric devices and can require careful setup, but people are successful with them.
I'm more of a tuning dial, coils inside sort of guy
No matter how fiddly they can be there's nothing like listening to radio waves with a good old valve receiver(!) It may be old fashioned but the clarity of reception has to be heard to be believed.
Most will never experience such a thing unfortunately.