USCCB website uses the NAB psalms. Other translations are also ok for use in the liturgy at the moment.

Are the NAB psalms, Grail Psalms, Revised Grail Psalms, and Abbey Psalms all approved for use at Mass?

The USCCB website does not use the current NAB psalms, but an older NAB translation instead. See this discussion:

As for the psalm verses in the Lectionary, they are taken from the NAB 1970 translation (which in turn is base on the Confraternity version. In fact, the text of all the OT readings in the Lectionary are taken from this translation, with some modifications here and there. See this link for more information.) This accounts for the version Xopheros mentions, "I rejoice because they said to me . . . " which is from the NAB 1970 translation.

Further, the responses (antiphons, refrains) to the psalms are translated from the Latin, which is the reason they do not always correspond to the same verse if it appears in the psalm verses. The translation of some of these antiphons was changed or modified for the 1998 Lectionary.

Abbey Psalms will be what is going in the new Liturgy of the Hours, the Catholic American Bible, and the next US Lectionary edition... but I think somewhere behind the scenes what is being prepared for liturgical use has some edits relative to the current published versions of that translation.

I think there also exists a textual variation in the responses that traces back to an ICEL version, but don't quote me on that.

GIRM 61 offers this:

In the Dioceses of the United States of America, instead of the Psalm assigned in the Lectionary, there may be sung either the Responsorial Gradual from the Graduale Romanum, or the Responsorial Psalm or the Alleluia Psalm from the Graduale Simplex, as described in these books, or an antiphon and Psalm from another collection of Psalms and antiphons, including Psalms arranged in metrical form, providing that they have been approved by the Conference of Bishops or the Diocesan Bishop. Songs or hymns may not be used in place of the Responsorial Psalm.

(Bolding is my own, as is the following comment)Note that such approval must be current, so if the local bishop back in 1970 approved something don't assume it necessarily remains approved, your diocese should have a worship, liturgy, or music office that can help clarify. Your pastor may also have preferences / directives... mine is pretty set on using the lectionary wherever possible so that it matches resources in the pews and materials the congregation access at home.

How and where would you discover that a text has been 'abrogated'?

Paul Inwood is almost always mad anyway.

Are there people working for the church, directly or indirectly, as lay ministers in music or liturgy that are not unhappy? In one form or another? (Joking... but not really joking.)

The text of the translation is what is primarily approved, so any of the approved psalm texts you could pair with a psalm tone or such like. Although you would need permission of the copyright holder to actually set the text to music.

Also we have the weirdness in the USA that the conference semi-officially delegated to each and every bishop the ability to approve music on behalf of the entire conference, which is plainly contrary to Rome's intention, but it is the state of things. Thus we have paraphrases of psalms that are set to music and approved by a bishop somewhere that in no way conform to the norms for translating the scriptures... are those still approved as psalm alternative or only as a song that can be used elsewhere at mass?... no one really knows. By that expedient we have for example some bishops approving English translation projects of the Roman Gradual, including thus the Graduals themselves... which is a super murky point as that sort of translation of a normative universal liturgical text is not really supposed to be done by an individual diocese on its own... but embrace the chaos I guess, if you feel like living dangerously.

As to abrogation, I doubt any diocese actually has clear organized record of what is and is not approved for use by prior bishops locally, at least not in any accessible way, perhaps a document misfiled in the archive somewhere. So, finding a psalm translation with long ago approval, I would not assume the intent of the current legislative authority in that diocese is that it remains approved, especially given the displeasure of such authority is sufficient to render you unemployed and unemployable in that diocese. Recalling also the bishop as the chief liturgist. Plus, any document declaring abrogation would be similarly hard to locate to check if it is still approved or not. So, best to check in advance with those in authority would be my advice.

If your goal is to get as many parishioners singing consider using the same resource provided to parishioners.

I have switched to the version on the USCCB website. I'm new at this parish, and at my last one they used the version in Worship 4, which is the Revised Grail Psalms.
If they put me in charge of the world, the Coverdale psalms would be allowed.

I've been migrating all of my psalm settings over to the AP&C so that the amount of editing I need to do once the lectionary officially drops is minimal. If you contact the copyright office, they will seed you a provisional copy of the psalter (basically with the current antiphons married to the AP&C verses). I stress that this is provisional, since the dust hasn't settled on the antiphon translations yet, but at least you could be normalizing to something closer to the forthcoming lectionary.

NB: at some point you'll have to rip the bandaid off anyway, because there will only be a short period (3 years, if memory serves) to transition, after which ALL old settings will be officially deprecated across the board. That was part of the USCCB's approval announcement last year. So they will have to get used to the new version, like it or not.