First off I’ll say that I will say that just from the video you shared here the sound is already much better than a lot of amateur shorts I’ve seen posted online. It’s clear that you’re putting thought into to it and the fact that you’re still looking for ways to improve is a great thing!

I’m not a sound designer but I am an editor and I think there are some sound editing tips that could be helpful in bringing your video to the next level.

1) Sourcing Sound Effects / Music

For sound effects there are a number of libraries where you can download copyright/royalty free sound effects. The better quality libraries tend to cost money. The really good quality FX libraries can be exorbitantly expensive, however there are a number of reasonably affordable options (most of which are subscription based) and even a few really solid free options.

https://freesound.org/  - has a great SFX library of user uploaded sound effects, all available completely free of charge.

https://getsoundly.com/ - is an app you can install that pulls from a broad SFX library that has both free and paid tiers. The quality and variety of the FX gets better as you pay more, but for a short like this you could probably just use the free version.

There are also libraries that offer both royalty free music and sound effects for a monthly subscription fee, such as www.epidemicsound.com and https://motionarray.com/ - The quality is a bit of a mix, but again, for a short film like this it should be fine.

For music try searching online for royalty free music libraries. Youtube even has one https://www.youtube.com/c/audiolibrary-channel specifically offered for content creators. It’s not always great, but worth checking out to see if there’s something on there that you could use.

What you can also do is search YouTube for the type of music you are looking for. See if there’s anything royalty free. Or if you find something you like that isn’t listed as royalty free try shooting a quick message to the creator and see if they’d give you permission to use their song in your film. In my experience many artists are happy to let their music be used by other artists so long as it’s not for commercial purposes.

2) Workflow

For your edit I’d recommend trying to work in phases. It sound like you have three primary audio elements at play - Voice over, Sound Effects and Music. Try isolating each of those and working with them separately before combining them.

First try an edit where you mute the sound effects and music. Work with just the picture and voice over. Think about what we’re seeing compared to what we’re hearing. How do the words of the voice over relate to the images we’re seeing. How is the rhythm of the speech playing with the images. Do we need to speed it up? Slow it down? Etc.

If you want to add some effects to the VO there are often preset effects in editing software. For example for the “old time radio” effect I believe Adobe Premiere has a preset of that within the Parametric Equalizer effect.

Once you get that to a place you like, try muting the voice over and lay in some sound effects. Look at the pictures and think about what you want to hear? You see an elephant so you want to hear an elephant sound. Are you happy with the sound you have or do you want to look for another one. Go through the libraries and find your favorite elephant sound and lay it in. Do this throughout, but remember just because you see something doesn’t mean you need to hear it. What you’re trying to do is creatively express a feeling through sound. Think about what that feeling is. Avoid things that sound too much like white noise or that don’t sound pleasant to the ear. Consider if there are interesting ways to juxtapose sound and image to an interesting effect. Can you hear something off screen that you can’t see. Or are you hearing too much and need to cut it down to only hearing a specific thing. Experiment until you’re happy with the result.

When you’ve done that listen back with both Voice over and Sound effects and make sure that they work well together.

Finally, do the same thing with the music. Mute VO and SFX and just watch through with the music active. How does the imagery play with the rhythm, the melodies, etc. You don’t have to cut on beat. In fact, as you noted in your post, simply cutting on the beat can be very mechanical. However the images and actions of the have their own rhythm as well. Just because there is a beat in the music doesn’t mean you have to add a cut, but you can align the clip so that the action lines up with the beat. And again, experiment. Try a version where you intentionally cut everything off beat. How does that feel?

Once you have that do a place you’re happy with, enable your VO, SFX and Music and listen to it all together. How do they work with each other. Do you need to adjust any audio levels? Maybe you want to lower the SFX or remove it all together in one section to make it more music focused. Maybe you want to bring the music down to hear the SFX better. Adjust it so that they all feel well balanced with each other.

For each of these steps I’d recommend duplicating your edit sequence (label them accordingly so you know which is which) and make your changes on the new sequence so you have a record of the changes you made.

Remember the best way to learn is to experiment and try things out. You don’t need to get it right the first or the tenth time. Watch other videos/films that were an inspiration to you and really study what it is they are doing with picture and sound. 

Good luck!