It may sound obvious, but the key to faultfinding is to methodically isolate parts of the signal chain to identify which part contains the problem.
In this case you already suspect that you might have speaker trouble, so a test you can do is try connecting your amp output to a different speaker cab and see if the fault clears. If it does it suggests you have a speaker fault, if not then it is back up the chain somewhere, say the amp itself or your signal source, guitar, stomps, leads.
Assuming the fault disappears when using a different speaker cab; apart from the cone being damaged there are a couple of other faults you can get in speakers, damaged voice coil (inside the magnet), and the braids that carry the signal from the speaker terminals across to the back of the moving cone.
A dislodged voice coil generally makes the speaker sound "choked" and thin, while a frayed braid can produce clean low levels but may "fart" when driven hard (which sounds like what you're experiencing).
A couple of test you can do (with the amp off) both involve moving the cone by hand, but this must be done gently, and with the fingers distributed around the cone so you don't distort its shape. The first is to press evenly around the central dust cover and listen for any scraping sound. It should move freely without any scraping, if not this is a sign that the voice coil may be coming apart and fouling the magnet gap (but if you distort the shape of the cone you may cause a good coil to scrape anyway).
The second test is to disconnect the speaker from the amp and clip an ohmmeter onto the speaker terminals and again gently move the cone forwards and backwards. The reading should remain quite steady, but if it shows any intermittancy you may have a frayed or broken braid. In some cases you may also get crackles and even loud buzzes or blurts. You can also try (gently) wiggling the braids, and again the reading should be stable and there should be no crackles or blurting.
A damaged voice coil means the speaker is basically stuffed and needs to be replaced or re-coned.
Damaged braids are easier; they almost always break just behind the speaker terminals, and if you look behind using a small mirror and light you may be able to see where it/they are frayed. The simple fix is to solder the frayed ends back together (the braids normally have a fabric core that holds them in alignment). Done carefully this can be an effective repair and give the speaker several more years of life. If you do find that you have a broken braid it is generally much easier to remove the speaker for repair than try to do it in place in the cab; just watch you don't damage the paper cone in the process. HTH.