It's probably that you need practice focusing on voice intonation and accuracy.
Is there a way to improve my pitch recognition for my own voice? Or is it just a matter of improving vocal accuracy?
I think it's both.
I'd try predictive ear training, which is awesome for your voice intonation, improvisation, and general musicality. The idea is to sing an interval, chord, or scale, and then play it in your instrument to hear how well you did. In your specific case, you can focus in the intervals and melodic scenarios that you are most struggling with (recording yourself is great way of figuring this out).
In the first levels you play a random note on your instrument, and sing all the intervals you can with that note as root. After you sing an interval, check your accuracy by playing that same interval in your instrument. With some practice you should be able to sing all intervals.
Then you move on to more interesting stuff. There's an exercise called "compete the chord" where you play the root and 5th on any instrument, and you sing the 3rd to complete a major or minor chord (and as always, check your accuracy after you sing). You can do this omitting any note from a chord, and the chord can be as complex as you like.
For scales, you should be able to sing all modes of the major scale. Again, listen to any random root, and sing a mode (or all modes!) with that note as root. You can make this as hard as you want to adapt to your level. Some people start by playing one note to introduce the root, and sing the Ionian mode both ways, then when you are back to the root go up one semitone and sing the next mode (Dorian) without listening to any outside reference, and continue until you run out of modes. The exercise ends with the root of the last mode (Locrian), you check on your instrument if that last note is what you sung.
You can adapt all these to your level and to any note dynamic you can think of. Play a note, then sing a triad with that note as root, or maybe the note you play on your instrument is the 5th of a chord now, or the 3rd of harmonic minor, or anything you can think of.
Some examples of predictive ear training by Rick Beato:
And a very useful comment from that video:
Opera singer here. Don't try to "hit the notes". Notes don't like to be hit. Instead, try singing notes with a strong dental consonnant onset and then, the vowel has nowhere to go but on the right note. "D" or "T", with feeling your tongue right behind your upper teeth, then the vowel can even be "uh", as in "Duh"! Also, a strong "B" or "P" onset can help. And I promise you won't sound "all opera", unless your body is meant to.
Some examples by Aimee Nolte (don't mind the title, it's actually about predictive ear training, but through a very interesting story about her experience with jazz scales as a teacher):
If you can though, consult experienced vocal coaches and teachers. Those people know what they are doing, and can do wonders for your voice.