Hoo boy, I think it's bad.

Looking at my previous work it feels like a dime-store novel when it comes to romance. It feels trashy in a way. Like, the uncomfortable kind of trashy. Romance isn't my forte, but I still need it for the story to go on. How can I improve that particular element?

Well, to quote Garnet:

“Love takes time, and love takes work.”

Kind of the same with stories.

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I think a big reason why you feel that way is because Romance Novels have an unfortunate stigma of being trashy.
It doesn't help that a lot of them have very poor pacing, unrealistic characters, and very rarely depict what a healthy relationship even looks like.

Look at 50 Shades Of Grey:
If you took away Christian's money, then the books would read like a season finale script for Criminal Minds.

That aside, practice until it feels right.

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Do you need help with romance itself? Try talking with your mom, or some other girl you trust to have such a talk with, about how girls generally feel about falling in love, being woo'd, that sort of thing. Or do you need help with how to portray romance in a literary style? Nothing like the classics. Put your story on hiatus for a bit and use the time to read Jane Austen (yes, all 6 novels...), even Shakespeare with "Romeo and Juliette" being the most obvious, but many of his plays have romance as a subplot as well, though I'd be careful of "Taming of the Shrew". And then there are the Sonnets...
You might even find some tips from reading Charlotte Brontë, though "Jane Eyre" might be more about how to depict love going wrong.

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"Taming of the Shew" if more of a story about using marriage for social/economic gain and a gentle intro to how Stolkholm Syndrome works, than it is a romantic piece

That's just my opinion though.
I've known people who believe that the dynamics in that play are nessicary for a healthy, functioning marriage.

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Yeah, I don't know why people think it's so romantic either.
Have you seen this review?
https://www.youtube.com/watch?v=sVbN3_mddZM

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I think that it was supposed to be one of Shakespeare's Comedies and humor was different back then

I have not but I'll make time for it

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Yep...sign of the times

Just imagine what the grandkids will be having gut wrenches about 50 years down the line