Visit a public bath.
Okay, you'll feel more like you fit in. Tiny difference. But trust me, a good, healthy mental state is extremely important for surviving your time in Japan.
(Disclaimer: if you have tattoos, you'll have trouble finding somewhere to soak. Many public baths don't let anyone with tattoos in, foreign or Japanese. This is changing, slowly - with the Olympics and Osaka Expo etc. in the near future, there's pressure on bathhouses to admit a more diverse crowd.)
Nobody will stare at you
If you happen to live in a less touristy part of Japan, or you're VERY visibly foreign, you'll get stared at. That's inevitable. Little kids will gawp, old ladies will side-eye. And you'll know it's happening, they're rarely subtle about it.
Being so openly ogled all the time isn't fun. It gets kinda annoying to put up with, and it can make you feel like you'll never be truly accepted here no matter what you do.
At a sento or onsen, everyone's naked. And that's not a big deal. You're not allowed to use your phone in the changing rooms, so there's no chance of being papped. Openly gawking at the other bathers is pretty rude.
There will be brief glances - is this bath getting too full? Is there room on that bench to have a sit down? Is that person getting out so I can take their spot? Nothing more.
Besides, you're all naked. Did I mention that? Nakedness is a great equaliser. Friends and family members often bathe together. Everyone looks different, and nobody gives a crap.
Awkward 'language barrier' moments are rare
If you're not great at speaking Japanese, or just not in the mood, public baths are actually a great place to go. There's very little reason for anyone to speak to you directly.
Every sento I've been to had a ticket vending machine at the entrance. Buy tickets for your entry, rental towels, toiletries, etc., and wordlessly hand them over at the desk. Well, as long as you know what the writing on each button means. Make a note of the right kanji and you won't have to ask.
There are often coin-operated massage chairs, leg massager machines, and arcade games in the communal spaces. Some of the restaurants have a touchscreen menu and ordering system.
You can make a 2-hour trip to a public bath, enjoy yourself, and not have to say a word (other than 'please' and 'thank you', which are simple enough).
I've only had an employee approach me while I was bathing once. And she used 2 syllables to get the message across: kami. 'Hair'. I'd forgotten to tie my hair back before getting in the water - another no-no. She quickly pulled an elastic band out of her apron pocket for me, and went on her way before I could even say 'sorry'.
You'll stop caring, too
The initial reluctance to strip off - and the initial shock of the cold shower on the way in - passes before you know it.
But I can tell you the exact moment you'll stop caring about fitting in here. It's when you first settle your bum into the water. Once that 'ahhhh' feeling kicks in, you're good. You won't want to hide. You'll want to stretch out, and let waves of fabulously scented hot water wash over you.
Like I say, it's more important that you feel like you fit in. The thought of being constantly watched and judged keeps you tense. A little naked bathing (with a variety of equally naked strangers) makes that stress go away. Give it a go sometime.