Is it permissible to send a message that includes an oath by Allah, commanding its forwarding to others, and a request for prayers for the sender's husband, while reminding the recipient that it is "a trust upon my neck until the Day of Judgment"?

Requesting prayers for a sick person is permissible, and it is commendable to fulfill an oath in this regard. If the matter pertains to praying for the questioner's husband, then she should do so. However, we do not recommend re-sending the message to a list of contacts due to the burden it may impose on people, unless she predominantly believes that some of the contacts on the list are concerned about the patient's condition, in which case she may send it to them. Her statement, "a trust on your neck," does not obligate the questioner to anything, for a person is only bound by oaths and trusts that they have themselves made obligatory or have accepted to bear.