You might say "a cheezy movie" meaning it's over-sentimental. I thought くさい映画 might work, but maybe that applies more to bad acting. How about くさいセリフの映画？ A word like 感傷的 seems a bit too formal to use for cheezy. Jisho suggests ド派手 but I feel like that doesn't capture the sometimes positive meaning that cheezy could have.
Any suggestions?
I think here is what you want, though spelling is a bit different.
I quote all what is written by a native speaker of English. The part in bold is what I gave and is not in the original text.
This is my attempt in English for convenience to the questioner, but since the actual text is written in Japanese, please read 日本語 Japanese when in doubt.
quote
This time, I would like to introduce the slang "cheesy".
At first glance "cheesy" seems to be related to cheese, but in fact it is not so, it means "cheap-looking" or "poor quality".
Especially "what seems desperate" and "what doesn't look like genuine" is expressed as "cheesy".
We often hear "cheesy" in everyday conversation, but it is somewhat difficult to explain because there are no similar words in Japanese.
And native speakers often use this slang when they want to explain such movies and music. For example, I don't mean to be rude, but Celine Dion is often said to be "cheesy."
I think that the movie Twilight is really cheesy.
トワイライトという映画はすごくクサいと思う。
My friend always wears a bow tie and a suit. It’s really cheesy.
友達はいつだってスーツと蝶ネクタイ姿なんだよ。本当に趣味が悪いだろ。
But cheesy does not necessarily have a bad meaning.
I really like cheesy, old movies. You know, like Home Alone and Ghostbusters 2.
僕はわざとらしくて古い映画がかなり好きなのだ。例えばそれは、ホームアローンとゴーストバスターズ 2なんかだ。
unquote
The questioner wrote that cheezy/cheesy also has a positive meaning, but since "わざとらしい" introduced in my original answer is not considered to be so positive as Japanese, I searched on the Internet for another translation and got a result. I'll show the result as follows.
The second highest rated definition of cheezy/cheesy in urban dictionary has the following expression.
Sentimental and/or dramatic, yet superficial and unconvincing.
Examining the first half of the above definition, I think that cheezy/cheesy is something a little closer to "演じる play a part" explained here, not "演じる" generally defined.
Since it is written in Japanese, I quote the full text and translate it into English and show it as follows.
"What does it mean to play a part?"
What is acting?
I think that the answer is different depending on the person. There is a story that I was taught on this theme when I was a student of an actor training center, so I will talk about it."What does it mean to play the part or play the role?"
To play the part is to tell a lie with your full strength.I was taught like this. Even if you have become an animated character thoroughly you are not the character himself/herself. Whatever role you get in the drama, you are not the person of the role but you are only playing the role. No matter how hard you may play, this is reality. Then what shall we do? There is nothing but to tell a lie. Of course, there are easy lies and so forth, but it comes out soon. So you have to tell lies with your full strength.
"Con artists are good actors"
Imagine a con artist. He will become a salesman, and the like, and cheat easy mark/victim. Why does the victim get caught? That's because the con artist can play the role of a salesman perfectly.
Let's practice to seriously lie from the bottom of your heart. However, it is allowed only on the play, please do not do it in everyday life.
With the same meaning as "演じる", there is also a way of saying "演技する". Though both words have almost the same meaning as "to play" or "to act", the impression of "演技する" differs greatly by the watcher. Even if you watch the play or act of the same player/actor, one person would badly evaluate the performer by saying "He/She is performing or acting deliberately, so I feel disgusted" while the other person would appreciate the same play/act of the performer highly by saying "He/She is acting very well".
次のところに cheezy とつづりが違いますが、スラングである cheesy についてのネイティブによる解説がありますのでそのまま引用して紹介します。但し、太字は私が付けたもので、原文にはありません。
【以下引用開始】
今回は、「cheesy」というスラングを紹介したいと思います。
一見「cheesy」はチーズに関係がありそうですが、実際はそうではなく、「安っぽい」や「質が良くない」などという意味になります。特に「必死になっているように見えるもの」や「本物っぽくないもの」は「cheesy」と表現されます。「cheesy」は日常会話でよく耳にしますが、日本語で似ている単語がないので説明するのは少々難しいです。そしてネイティブは、よくそのような映画や音楽などを説明したい時にこのスラングを使います。例えば、失礼な話ですがCeline Dionはよく「cheesy」と言われています。日本の歌手だと、、、それは皆さんの価値観次第ですね。
I think that the movie Twilight is really cheesy.
トワイライトという映画はすごくクサいと思う。
My friend always wears a bow tie and a suit. It’s really cheesy.
友達はいつだってスーツと蝶ネクタイ姿なんだよ。本当に趣味が悪いだろ。
しかしcheesyは、必ずしも悪い意味になるわけではありません。
I really like cheesy, old movies. You know, like Home Alone and Ghostbusters 2.
僕はわざとらしくて古い映画がかなり好きなのだ。例えばそれは、ホームアローンとゴーストバスターズ 2なんかだ。
【引用終わり】
質問者が cheezy/cheesy は悪い意味だけではないと書いておりますが、私の最初の回答で紹介した「わざとらしい」は、日本語としてそれほど悪い意味が拭（ぬぐ）われているとは思われませんので、別の訳を探しました。その結果は以下の通りです。
urban dictionary の２番目の定義に次のような表現があります。
Sentimental and/or dramatic, yet superficial and unconvincing.
上記の定義の前半を見ておりますと、cheezy/cheesyは、一般に定義された「演じる」ではなく、ここに説明のある「演じる」と少し近い意味ではないかと思います。
さわりの部分を引用しますと、
「演じるとは、全力で嘘をつくこと」とあります。
同様の意味では「演技する」とも言うように思います。特に「演技する」は、見る人によって感想がわかれるもので、「わざとらしく演技しているのでいやだ。鼻につく。」と悪い評価をする人と、「上手に役を演じている/演技している」と高い評価をする人とがいるように思います。
cheesyon ALC.