Restrictions build creativity. What if a species cannot communicate with more than a single syllable at a time. Maybe everything must rhyme, or be fixed number of words per sentence. Some languages every other letter must be a vowel (Hawaiian). There can also be tonal properties, consider free adjectives but with different meaning dependent on the sound of delivery.
The best way is to listen to other people talk more with an ear to how it would feel to write that. Get out of your own head a little. Dialogue writing is non trivial, and a very worthwhile skill to spend time on. Do impressions of people unlike you, roleplay your characters, spend a while as someone else.
Take examples from other languages. Hawaiian has a much smaller number of phonemes, while Mandarin uses additional tones on top of consonants and vowels to create different words. Khoisan languages use clicks and stops as consonants.
Sign language is entirely visual while being just as expressive and emotive. Greek has four different words for "love" but colloquially uses the same word to refer to a foot and a leg. Classical Hebrew omits vowels from written words, as the vowels are implied.
Tamarians speak entirely in metaphors, and their language is impossible to translate without an understanding of their historical references and memes. Sith is a forbidden language that translators recognize but are restricted from translating. Prawn language from District 9 is entirely made of clicks and squeals. Chewbacca, R2D2, Huttese, all were unrecognizable as languages but characters just understood. If you haven't seen the Arrival, it's an entire movie about xeno linguistics and different kids of language.
The main question is whether you intend for them to be understood, or if the language is just intended to be perceived and not understood? What is the effect you want? Do you want it to feel completely foreign? Or does the language serve some other narrative purpose?
