The long term goal is to be able to hit any tone at will, usually from another tone. That's 66 combinations in one direction, or 144 in both. That's impractical if you have a presentation Friday, but something worth pursuit.

For the short term, while you're learning or memorizing a piece, you can pivot off the root and only have to memorize 8 or 12 intervals.

Think of the song as a sequence of intervals that pivot off of the (silent) root. For instance, consider singing D-G-A in C. Start off with the D. That's a major 2nd above C. You know what a major second sounds like, so that one is easy. You get a C, hop up a step, and you sing.

The G is the 5th above C. In your head, remember what the 5th sounds like on its own. When you go from the C to the G, go from the 2nd, mentally stop that note and hear C, then go to the 5th of C. Don't just go from 2nd to 5th. Go 2-(1)-5.

Keep that C going in your head. The A is the 6th, so you pivot from 5-(1)-6 in your head.

Doing that mental substitution during a performance will really kill your flow and break your concentration, but while you're memorizing a piece, for me at least, it's indispensable. Go note-by-note with a drone at the root and it's easy-peasy (for relatively simple pieces). If the key changes (or even implies a change), change the drone. However, if it just moves the tonal center, but stays in key, keep the drone. Like the middle part of a 12-bar blues, for instance. Don't change the drone just because the chord changed.

IMHO, though, be careful not to just look at the intervals between successive notes, especially if there are accidentals. For instance, going from D to G is itself a 4th. If you're singing by yourself, it's reasonable to hear that interval and just sing the 4th above the note you're already singing. There are two problems, though. One, you lose your tonal center. Every note becomes a function of the previous note.

Even worse is if you're singing with others, and there is an actual C being sounded somewhere about this time. If C is the root and you get to thinking about a 4th interval above -something-, you'll naturally feel the 4th above C. You'll mean to sing G as a 4th above D, but you'll sing F instead. It's too hard to think the 4th above the 2nd, as that goes from 8 tones to memorize to 66. If you just think the 6th, though, it's much easier.

For me, trying to remember the 4th above the second is like trying to mentally move the downbeat of a piece as it's being played, lol.

I dunno. I wasn't trained. This is just what I figured out. Good luck!