[before this discussion begins, I want to clarify that I am not an otaku who has unrealistic expectations of a society. Thank you.]
I am engaging with more Japanese content online to learn more about the way people act, from their gestures to their speech so I can integrate better if I ever decide to visit, or meet someone from there. I realized that people there speak like....actual people. Not like your average anime protagonist. I didn't know what I expected honestly, but since I only learned how to speak by listening to seiyuus, my voice is unrealistically high pitched and bubbly. I feel like this would sound very odd and ridiculous if I ever spoke to a native. How would this affect me? Is there a way I can speak like a real person and not an anime girl?
Listen to podcasts, Japanese vlogs, converse with natives, etc... You can still watch animé and you can and still will learn Japanese with it to a certain extent, but since animé tend to use very casual speech and/or end of speech particles like ぜ/ぞ that are quite strong and used very often in animé it might make you sound a bit odd when you speak.
The better you get at Japanese, the easier it becomes to recognise what's natural and what's "anime speech". You also simply can't reach a high (or even moderate) level of proficiency from anime alone, so there's pretty much zero risk of being both fluent and speaking like an anime character. Any bad habits you pick up from anime tend to correct themselves naturally as you keep interacting with the language.
Anime also covers a wide range of speaking styles, and depending on what you're watching, characters may already be speaking fairly close to real-life Japanese.
By the way, I also think it's worth branching out to other media :) Particularly podcasts, vlogs, and livestreams.
PS: I've watched hundreds of shows over the past few decades, and I've never had any issues. Although I also feel like anime didn't help me with conversational skills at all XD
Even dramas and movies are exactly that, dramas. They don't do "everyday", which is, well, 99% of what we experience.
When I was a teacher here in Japan, I had a foreign student who was a huge anime fan. You could tell immediately that he borrowed heavily from phrases and intonation in anime, and I have to admit that it did draw some smirks and laughter from the native students.
Try to interact with more native Japanese, anime may be nice but other movies or series with native Japanese is great. I don’t expect every anime to be high pitched or bubbly though, as I am a great fan of the Japanese anime Detective Conan, and there is nothing wrong with my voice. I also find listening to simple Japanese audiobooks are great too. For audiobooks I recommend Japarrot on YouTube.
Even within anime, there's a pretty big difference between the kinds of speech used in a battle shounen, a fantasy isekai, a highschool romcom, or a seinen drama. So when you say "anime accent", that could mean anything from Jotaro Joestar to Bocchi to Makio Koudai.
In general, the best thing you can do is diversify your input. Obviously that means watching more than just anime, as others have noted, but even when you do watch anime try watching more genres too.
You can watch some interviews or anime promotion streams with the same seiyuu so you can feel the difference between their anime voice and how they actually speak.
There are some words in anime that everyday people just don't use particularly 俺様(おれさま) which is an extremely informal way to say one's self and no one uses it in everyday life.
Also, tones in animes are very exaggerated in tone and tend to use informal Japanese speech. It's a bit hard to explain but think of your faveroite English cartoon or movie or drama. The way they speak is very exaggerated but you don't exaggerate your tone when you speak normally to your family, friends, co-workers and such.
Sure, go ahead and learn some Japanese words, phrases or practice listening. I'm happy to admit I sometimes do that since I have no one to talk to. It never hurts to watch anime. Japanese movies and dramas but keep in mind everyday Japanese people won't exaggerate their tone that much as we would not exaggerate in our everyday lives.
if you don't look japanese, or if it's evident in some other way that you're foreign (which it will be as soon as you begin speaking), then having "odd" japanese imho won't be a big deal. caveat is if the type of japanese u are using is especially crass, while also being highly fluent, at which point your interlocutor might think you are being intentionally rude or hostile toward them.
as for obtaining a more natural style, that will come automatically through immersion in a japanese environment. even ppl from different regions of japan and social classes or work environments or hobbies and interest groups within japan have different diction, pitch accent patterns, gestures, behavior...
i wouldnt worry about it. most people can figure out social cues when in a different environment. if ur speaking too loudly or in a voice that is very different from natives, you (iiif you are like most people) will realize that and correct automatically. dont worry too much about being "odd" at first brush... you're going to be "odd" anyway because you are a foreigner. your language and even sometimes things like behavior, dress style, and other things will blend into reflecting your new environment over time. listen to others, and pick up on social cues from others. people are remarkably adaptable if you are open minded.
