Every interval has a specific sound - but everyone of us may perceive that differently. But you can build some personalized mnemonic aid especially for you. Custom as it were ;-)
I'll tell you some of mine just to give you an idea of how this could work for you.
I'll start off with the b7th: Leonard Bernstein - my hero, as conductor and composer
-> 'Somewhere' from 'West Side Story': (Maria/Tony:) There's a place for us...
There's (1st) a (b7th) place (6th) for (4th) us (2nd) -> only the first two notes matter. They are part of a dominant-7th-chord (mixolydian scale)
A difficult one #4th: Ennio Morricone - my super-hero, as a composer for film-music
-> 'Once Upon a Time in the West': Orchestral part underneath the famous guitar melody and all over the place as quasi Leit-Motiv...
A - D# - E - A - D# - E - D# - E -> You will never forget this - definitely get the movie...
b6th: again Ennio Morricone
-> 'Cockey's Theme' from -> 'Once Upon a Time in America': the Panflute playing this unforgettable melody...
G - Eb - C - D - Eb - C - D - Eb - C - D - Eb - C - D - D - C - D
Now go and find your own ones for the 3rds - I didn't want to influence you on these ones. If you find them on your own the mnemonic effect will be much stronger because it's your choice.
Once you have found your favorite melodies and mnemonic helper you will never ever again miss a beat ;-) Never forget - it's all about music, listening and recognition. Systematic theory comes second - but it will, definitely...
Happy listening!