As mentioned in the stickied message in this forum, a recording would help, or a screen shot. Fortunately, I can take a good guess as to what this is, because other listeners have made the same mistake of thinking this is CW. Without a recording or a screen shot, or both, I can not be 100% sure, but I would bet a sizable amount I am right. Assuming I am, the following applies:
This is not the letter E, nor is it CW. Everything that goes beep and boop is not CW or Morse code

Many digital modes have a pilot tone. A pilot tone can be intermittent or it can be on the full time the digital mode is on the air. When they are intermittent they typically occur at the beginning of the transmission. That is what you have here, a pilot tone at the beginning of a digital transmission, in this case HFDL (High Frequency Data Link), also called HF ACARS.
HFDL is a data link used by commercial aircraft to report various information to ground stations. It consist of a short pilot tone of 250 msec (0.25 second) sent on 1440 Hz (when properly tuned in USB) followed by a 295 msec 2PSK preamble and then 1.8 or 4.2 seconds of 2PSK data.
The 1440 Hz pilot tone makes what you are hearing as the letter E and the PSK data is the whooshing noise you hear. The multiple power stages or levels you are hearing are the different received strengths of the aircraft reporting.
The frequencies you report are close, but no cigar. Try USB mode, and then try 21928 kHz, 21937 kHz, and 21949 kHz. This should result in the pilot tone, what you are taking as the Morse letter E, at about 1440 Hz audio freq, followed by the data.
There are several possibilities for who the sources are. Both 21928 and 21937 kHz is used by Molokai, Hawaii, and Barrow, Alaska and aircraft reporting to those stations. 21949 kHz is used by Auckland New Zealand, Hat Yai, Thailand, and Johannesburg, South Africa, and aircraft reporting to those stations.
Later I will try to put up a recording of HFDL on my Youtube channel, so you can compare it to what you are hearing. It may not happen for a couple days though.
T!