I feel like Grammarly is more suited to things like writing letters, emails, product descriptions and essays, where "correct" English written with clarity will help the writer come across clearly and so sound professional.
But of course, most native English speakers don't use proper grammar when speaking, regardless of which English speaking country we're from (in fact some might say we English people are the worst offenders!)
When looking at some of the phrases in the first episode of your comic, there are some I'd rephrase to sound more like natural spoken English, like:
"I thought we were fine in the last 3 months?"
I'd probably change to:
"I thought we were fine this last three months?"
But generally it's not bad!
Reading stuff in English and maybe watching recent films or series is a good idea for developing your skills. Although, if you do, make sure to note which country different films and series are set in. American English and British English can be quite different, so learning the differences between the two or just focusing on learning one first tends to be better than trying to avoid all specific grammar and slang. You can write a comic in perfectly neutral international business English, it just means missing out on a lot of flavour, so picking one out of British or American English is advisable. "Hi! How are you?", is fine, but being able to use: "Hiya! You alright?" (British) or "'Hey! How's it goin'?" (American) when you need them is when the English really starts to feel real. When somebody knows how to use "bad" English the way native English speakers do, that's when you know they're really good at English!
(In case anyone needs this information: Canadian English uses British spellings but the way they speak is much closer to American, while Australian and especially New Zealand are closer to British. Most Fantasy films like Lord of the Rings lean closer to formal British, and Star Wars is... er.... all over the place; do not attempt to learn natural English speech from Star Wars. Other countries like Ireland and India have their own distinctive English, both are most similar to British English but with their own distinct grammar and vocabulary.).