I'd really like to improve my ear as I think it's one of my weakest areas. However where I practice everyone can hear me and am sure they don't want to hear me attempt singing, also I get super embarrassed.
Is there any other methods that are just as good that doesn't involve any singing?

There's an app called Functional Ear Trainer that I really like. It plays a three chord cadence in the key you're in before playing a note and having you name the interval. When you get it right, it walks down the scale so you can hear the relationship of the notes. I have it for android and I think there's iPhone version (it's free).
singing is the fastest and most reliable way to improve your ears, because it forces you to internalize the sounds to achieve a result (i.e. sing the note or notes).

it's easy to say "don't worry too much about what people will think", but something else that may help you is that you don't need to be singing like a professional. to train your ear, the main objective is simply to internalize and produce the pitches. I recommend humming if volume is a concern. if you're not aspiring to be a singer, there are ways around it. try different things and see what works for you.

but don't do yourself the disservice of training your ear without singing. it not only involves more work, but it produces fewer results.
Anfangen ist leicht, Beharren eine Kunst.