Singing can help. I didn’t have the confidence to try this/ thought I couldn't for years - and wish I had just tried sooner.
Using a tuning app to help you see what you are singing is a good idea. (You may be latching onto overtones or other bits of notes - a guitar sound may be easier to match than a piano - a note from an instrument will often for example contain bits of other pitches and you need to learn which is the main bit to pay attention to).
Some start by singing any comfortable sound, then using an app to tell them what the pitch is, and then using some kind of instrument or app to match that pitch - obviously if you have someone who can sing your note for you, all the better.
Some of us haven't had the experience of being in tune with someone else and need the experience of same and different. (Somewhere there's an odd story of a girl who learned to match pitch after realising what she thought was being out of pitch actually was the opposite.) I also found it helpful to hear a sound a major third away from me (Five notes up or down on the piano, counting top and bottom).
If like me you have never sung, you may just need to spend some time getting used to making random sounds and start with a small range - eg making sweeping sirens of sound up and down, imitate a fire engine. Or express surprise or disappointment- your speaking voice probably changes pitch. See it with the tuning app. Then try turning that into singing.
Some find humming easier - others recommend singing on the syllable 'bum'. Learning how to breathe for singing is said to help with pitch.
Learn to listen and try to remember what you have heard, and imagine singing the note before you sing it. But don't get hung up on it as you start.
If you can find a find-your-voice style choir/teacher, you may find it is full of people in a similar position and teachers with experience in helping. Beginner's Kodaly or Dalcroze classes are also designed for this kind of thing.
Are you really hearing yourself? Try cupping your ears/ recording the sounds you make - what comes out may not be what you imagine.
When you are close but off pitch, the dissonance can be quite crunchy - can you hear that? It can be easier to hear than being further off - again, if you have something that tells you what pitch you are singing, experiment with sounds around that note.
Spending time with really simple children's songs etc is also helpful - though be wary that you may be hearing an ingrained misheard version rather than the real thing) - and I still often find my poor singing gets in the way of perception sometimes.
When I started out I didn't know anything about relative pitch or harmony. (I didn't know for example that if you play the first and third note of a scale together, it can sound nice, and scales aren't random). Knowing about this helps. Plenty of online resources for that, eg - Andy Mullen used to have some helpful free resources - useful even though they went a bit fast for me - (https://theimprovingmusician.com/about/) - he's a follower of Edwin Gordon who did a lot of research on how kids learn music with a focus on trying to develop audiation skills - the ability to hear musical patterns internally)
Inventing tunes using a really few pitches - just messing about - is also a good thing to do.
Learn songs by heart - and when you can playing it in different keys. Edwin Gordon published collections of pattern-based songs for children. (Marilyn Lowe published a Gordon-influenced set of piano course books which I have found helpful.) Sing these into a pitch app once you can. (Eg singscope; or try use the sightsinging apps with their built in tunes for the feedback https://sightsinging.mystrikingly.com/ - you can start by playing yourself the sounds you are trying to match, sight singing can come later)
As your hearing improves, your memory for music should improve too.
You already have advice about drones etc
Persistence, patience, a little every day.
Start with recognising big differences.
Interval recognition is hard for some of us - as others have said, functional ear trainer's approach is more accessible- though beyond the basics you are likely to find it overwhelming at first.
Some intervals are thought of as consonant (Eg octave/fifth; some dissonant (Eg minor second, tritone). If you set an interval trainer to play harmonic intervals, can you hear if it sounds more or less settled?
Also there are apps that just ask you - higher or lower - someone else may remember the name of the app that does this with closer and closer sounds - the developer noted stringed instrument players before tuning apps were said to get good at discriminating pitch because they had the constant practice of tuning.
I do ear training exercises without singing too - my poor singing can get in the way sometimes and doesn’t always help me).
I can't claim any great success with these approaches and doubt I will ever play by ear - but they have made a difference to the way I hear.
There's a culture of perfectionism around music - being tolerant of what you can do is important. You learn by doing. While you are learning what you are doing won't be perfect. Especially if you are going to try singing, you need to be willing just to give it a go.
Also just relax and listen to individual sounds and patterns.
I also think that for me, the part of my brain that processes music isn’t well connected to the thinking/speaking parts of my mind - there is something about just letting go and trusting yourself to do that I still find hard. Interesting in this connection the reports about musical memory surviving in some people with advanced dementia.
There are lots of other solfege singing videos out there - find one you like - you can sing along as best you can without worrying if you might be out of tune - you are still listening and hearing, and your body is getting used to making sounds - Andrew Rostas has some I like
You can also look at primary school music teacher training videos online - the exercises they are learning to teach can be useful.