There are some FB EL84 amps about, Laney for one, and some folks don't like them. (If I were a cynical type I would say it is a cheap way to get a 20W amp!).
You should give an HT-20 a fair go, that has both biasing regimes.
Then, cathode biasing tends to go with thermionic rectifiers (hateful things) thus the "sag". You might find a CB amp with silicon derived HT a bit different?