[FrankHudson]

The OP seems to speaking of a number of issues, all around recording vocals.

Like Ray upthread, I'm not much of a singer and particularly for myself I have the "can't carry a tune in a bucket" syndrome. Also like Ray, I've gotten over the shock of hearing my recording voice after years of experience with it, and on good days I can control "red light fever" when recording as I'm often alone when doing it--but the pitch issues still mar the results.

A few things that may help.

Tweaking how you hear yourself when recording (monitoring) Nothing in this realm has helped my pitch issues, but all are worth trying. One can just use headphones on both ears, but some singers prefer twisting the set and using only one cup on one ear. I sometimes record without headphones at all when recording with acoustic guitar, which seems to help me a little bit, but if in fairly dead space treated for recording, there may not be enough information reaching my ear. Though I haven't been able to repeat it, during one produced recording the producer had me singing facing a pair of monitor speakers (the speakers were in the null space of a cardioid mic).

Temporary, just for the recording pass, modification of what you're hearing in the headphones etc. Upping an instrument or other vocalist(s) in the backing track that best helps guide the vocal over where it will sit in the "real mix" is one tack. Some singers may want "more me" to help hear what their voice is doing while tracking. Some singers are helped by having a bit of reverb to spruce up their voice while tracking, others may hear pitch issues better monitoring dry.

You don't have to nail it first take. Nowadays with home studios and digital recording, there's no reason to treat a pass at a vocal as a high-stakes, game on the line deal. Within the limits of your voice's endurance, you can always do another take. Modern DAWs make "comping" vocals (using the best parts of several takes knitted together) something to consider. One thing I've noticed, it it's good to do at least one take after you think you've done your best, "just for grins." That take after you think you've checked off the task of getting an acceptable performance may well be more relaxed and better yet.

Consider tracking the final vocals as a vocals-only pass--or not. Doing vocals only recording passes sometimes helps me, and I should take the time and trouble to do it more often. To some degree I enjoy interacting between my voice and my guitar playing, and many good players can walk, chew gum, play hot licks, and sing like a bird at the same time. Some players sing better when it's a more offhand task as they play their instrument. But that's not everyone.

If that voice you hear on playback doesn't sound like you, that could be a feature not a bug. Consider that voice a character you didn't know was inside you. Especially as this post is in the play and write section here this should be considered. Singers and writers both are said to have to "find their voice" and it may not be what they thought they'd find.