Hi, Mike. Welcome to the hobby.
Frequency drift is typical of radio equipment that is poorly designed, or has low-quality or defective components. For instance, old tube-based radios that also lacked advanced anti-drift or frequency stablizing circuits would drift as the radio warmed up. The heat caused changed in the components, resulting in the change of frequency. In transister radio curcuits like in those cheap 'walk around' radios of the 1970's, the small plastic and foil tuning capacitors inside would be prone to frequency drift because the 'Q' of the tuning curcuit was so sensitive. A very tiny bump of the physical 'setting' of the foil 'plates' would cause huge frequency changes. Often, in these low-budget radios, even bringing your hand or body close to the tuning curcuit would result in a shift in frequency. So, temperature, slight movements, stray capacitance: all of these would result in frequency changes.
More expensive radios were designed with higher-quality components, and had cuircuits that kept the temperature at a constant level, shielded the tuning curcuit from stray capacitance, and so on.
Newer digital radios used techniques that avoided these mechanical problems. So, it became possible to create low-cost digitally-tuned radios with a better control over such functions as the tuning of signals.
What does all that mean? There are radios now, within the budget of 200 to 400 that will work rather well for you.
For instance, here is a radio that fits your criteria:
Sony ICF-SW7600GR AM/FM Shortwave World Band Receiver with Single Side Band ReceptionDescription:The Sony ICF-SW7600GR world band receiver radio will keep you connected to FM, AM, shortwave and longwave stations with a PLL digital tuner. With single side band (SSB) reception, the radio will receive morse code, ham radio transmissions, and other types of shortwave radio transmissions that expand your listening experience beyond AM and FM broadcasts. To further enhance listening pleasure, the synchronous detection will reduce fading and regular beats.
It certainly is within your price range of $200 to $300.
There are others like that, but this is a good starting portable radio.
As far as buying used -- well, buyer beware. Like anything else, you must be very careful to know as much as you can about the radio before buying it. What is the ownership history? What condition is it REALLY in? Etc.
Finally: a bigger antenna can sometimes overload low-end radios. This example radio would benefit from a more 'proper' and larger external antenna.
I hope that my answer is helpful.