The document for the EF which has these instructions is De Musica Sacra (1958).

There are a couple of translations out there online of the whole thing.

However, the relevant bit is cited by the Liber Usualis, HERE.

Regarding your Adoremus with continuo, come to think of it, it sounds like you could do it at benediction, because benediction of the Blessed Sacrament is explicitly exempted from the prohibitions.

Incidentally, recently, although we usually sing O salutaris, Tantum ergo, and Adoremus in aeternum at benediction in their most typical chant melodies, unaccompanied, I have been making a point of doing the metrical hymn-style O salutaris and Tantum ergo when it is Advent or Lent, so as to bring out the fact that the organ can and is being used at benediction at these times. The resulting contrast seems good, and has also the advantage of giving out that the chant is the more festive.

What if vespers is prayed in the context of adoration? Would the exception of being able to use the organ during adoration supersede the prohibition of using the organ for liturgies?

Benediction is a solemn liturgical act (adoration on the other hand is devotional), and it can be fully joined to the liturgy of vespers such that the two are celebrated as a single liturgical act. It is specifically provided for in the relevant books, which I don't happen to have at hand.

80. Quoniam organi et magis quoque aliorum instrumentorum sonus ornamentum constituit sacrae Liturgiae, usus idcirco eorundum instrumentorum temperandus est secundum gradum laetitiae, qua singulis dies vel tempora liturgica distinguuntur.

"Since organ music, and still more that of other instruments, constitutes and adornment of the liturgy, the use of these instruments must accord with the degree of rejoicing that belongs to each day and to each liturgical season."

This is the general explanation given by De Musica Sacra, that the use of organ is an adornment which should agree to the level of rejoicing.

Right off, the exception is made for benediction, and later other exceptions are made for certain days and circumstances where there is indeed more rejoicing.

However, the only exception made for practical purposes is where it says:

83. c) Item organi tantum aut harmonii sonus permittitur in Missa et in Vesperis, solummodo ad cantum sustentandum.

The music of the organ or harmonium is also allowed at Mass and Vespers solely to support the singing.

Which to me says that if you were to use the organ in such fashion, it is not and ought not to be made to be an adornment of the liturgy; rather, it is in this case providing merely the crutch of utility music ad cantum sustentandum.

I also find it interesting that this crutch is only permitted in Missa et in Vesperis ; which I would argue means that in such times as these prohibitions apply, one can't use organ at all for vernacular hymns before or after high Mass, due to the fact that 1) the exception is only made for Mass and 2) everyone who is singing vernacular hymns before or after high Mass is able to do so for the very reason that they are outside of Mass.

So, my opinion is that this exception for practical purposes is made in promotion of the singing of Mass and Vespers, even by those who might need a crutch; also implying that if you are trying to sing other hours of the office, you are either biting off more than you can chew, or are demonstrating that you really do have the level of competence where you ought to be able to sing a cappella, like it or not.

[Pardon the digression; and I think I have said some of this before.]

Richard Mix - De musica sacra:

63. In addition to the classic organ, the use of that instrument called the "harmonium", is also permitted, but only on condition that its tonal quality and amplitude of sound makes it suitable to sacred use.

Example of pious exercise would be, mainly, Stations of the Cross?

Or are you suggesting that processional and recessional hymns would fall into this category?

But then, are the processions in and out "liturgical actions"?

Well, organ can accompany chant, and you can give incipits; I even heard this on Good Friday for the Passion, which was an occasion where a discreet pitch pipe would be better. But for Exposition you could have music. That being said, Benediction usually follows Vespers in the traditional rite, because the rubrics are insanely complicated.

I think that were one to ask Rome, the rubrics would also apply to Lauds: morning voice is a real problem, and it helps to have organ to give the full incipit of the psalm and antiphon. If not, then use a pitch pipe, and have someone hum the incipits to the cantors.