Should I replace these resistors on my vintage radio?
Carbon resistors like what's in your vintage radio are not stable over long time periods. In my experience, the resistance moves upwards. I have noticed this even on early Delco Studebaker solid state car radios that persisted with carbon resistors. Your radio probably has pre-WW2 resistors because WW2 made designers adhere to the preferred range which 250K is not. Due to the age of your radio, I think you should replace all the old resistors. Remember that many small modern resistors do not have the voltage rating. It is good to get the chassis to spec so the valves will all be at their correct bias points.
Your resistors, as you now know, are very old. Their manufacture was known as "composite". A crude simile would be an insulating "pipe" filled with the carbon composition and you stick the wires at the ends (or the caps with the wires on as I see in your case). They are prone to going high in resistance and well out of tolerance. If they carry current with a biggish voltage drop across them, even more so. Strongly recommend you replace them and keep your old valves working within their voltage tolerances.
If you are getting distortion in the audio, certainly replace the resistor which goes to the anode of the first AF amplifier. Typically that resistor was a "half-watt" one which typically would go high due to the "duress" it worked under. Put a 1watt or even 2watt resistor in place of that one. You would also do well to replace the resistor(s) which feed voltage to the screen grid of the AF output valve. Even if the screen grid is fed by voltage divider resistors which have gone high in value by equal amounts, I suggest replace to be prudent. The dynamics of the screen current through one of the old too-high resistors could screw your screen bias and also promote audio distortion.
Old composite resistors can also become noisy, so any crackling or "fried egg" noise and you should target not only those resistors above, but also the resistors at the control grid ("grid 1") of the (typically) two audio amplifier valves.
"Newer" (post-WWII) resistors are not made physically the same way, and are more likely to remain "quiet" as well as hold their value.
I'll try to "sneak" this one past in the guise of answering your resistor question. Check the DC voltage across the resistor which goes between control grid and "ground" on the output valve. Assuming the output valve uses cathode bias, there should be absolutely no voltage across that resistor. If there is, then replace that resistor as I already suggested, but the main thing is, the presence of voltage on the output valve control grid resistor when the output valve is using cathode bias, means that the coupling capacitor feeding audio from the previous AF voltage amplifier is faulty and leaking some of the HT voltage on to the grid, and this can also create audio distortion.
But keeping to resistors - while you are replacing the AF stages resistors I specifically mentioned earlier, that possibly capacitor is linked to those resistors so replace it too and then you won't need to eventually ask the next question of "where do I get another output valve?"
Good luck with your efforts.