No, they do not. Received transactions working fine; sending a transaction via electrum.dash.org results in the same problem described above by alexne.
Well and good if one has a Trezor. Not a solution for those who simply use electrum-dash, because it requires that a Trezor device be connected in the first instance. Thanks for the thought, but it doesn't help here.
It appears that I shall have to follow the same recovery procedures as alexne. As ldw says above, this makes for a terrible user experience. I'm afraid I would go further: it severely damages any notion I held of Dash as a mature, stable cryptocurrency. Thankfully my holdings in Dash at the moment are only small, a couple of hundred dollars at most, but rushing out 12.1 without testing has, for the time being, turned Dash from an asset to a liability. The only upside is that I feel much better about having previously disposed of ten times my current holding at around 20% less than the current market price, precisely because I didn't want all eggs in one basket.
Even assuming that I have access to a machine capable of running Dash Core and to a connection over which I can download several gigabytes of blockchain (which I do, but didn't when I opted for electrum-dash), the cost in time and hassle is substantial and the reputational cost to Dash and its developers is likewise. Given that a majority of electrum-dash users will presumably have chosen it because the available hardware or connectivity did not favour running Dash Core and many will not readily be able to work around the foul-up, I suggest that it may be a better idea to shut off the electrum servers and to break electrum completely until it can be fixed completely than to limp along in the present state.
As it stands, the user of electrum-dash via electrum.dash.org gets the semblance that all is well until he tries to make a payment, whereupon he has the frustration and embarrassment of being unable to do so, the anxieties of establishing what has happened and how to recover access to his funds, the hassle of the associated technical work and the impression of Dash and its development team that logically results. This strikes me as a worse outcome than overtly breaking it and overtly fixing it, or indeed than rolling back the embarrassingly flawed "upgrade" with another hard fork until it can be implemented properly.