It is an adjective. It has two meanings and they are always used in a negative way.
The first meaning is “to be heavy“. It is used to describe the taste/smell of something that is too strong.
The second meaning is “to be insistent“. It is used to describe an insistent person/thing.
Ken is asking Leo about his favorite dishes.
- Ken: I want to recommend delicious Japanese food to my colleagues from the US. What is your favorite one?
アメリカから来た同僚に美味しい日本料理を勧めたい。レオ、何が好き？ - Leo: I like tonkotsu ramen. The broth is rich and tasty.
豚骨ラーメン。濃厚で美味しい。 - Ken: It’s greasy and heavy, isn’t it?
脂っこくて、しつこくない？ - Leo: Not really. Once they have a bite, they will get addicted.
そんな事ないよ。一口食べたら、病みつき(*)になるよ。
Meg had an argument with her boyfriend last night. He is calling her to say sorry again and again, but she is ignoring him.
- Amy: Is that your phone I hear ringing?
メグの電話、鳴ってない？ - Meg: Yeah. He won’t stop calling me. So annoying!
うん。彼、何度も(電話を)かけてくるの。あー、しつこい！
Amy is complaining about her mother’s advice.
- Meg: How was your visit to your parents?
実家、どうだった？ - Amy: It sucked! My mom was on my case about getting married.
最悪！結婚しろって、お母さんがしつこかった。 - Meg: Really? You’re still 27 years old. Why is your mother pushing you to get married?
えっ？まだ27歳じゃん。なんで、そんなにしつこいの？ - Amy: I don’t know why, but she is excited to get grandchildren.
よく分からないけど、孫が見たいんだよ。
Ken is worried about Leo because he is coughing all the time.
- Ken: Are you okay?
大丈夫？ - Leo: I went to the doctor and took some medicine, but it seems like it doesn’t work at all. (sight) This cold won’t go away.
病院行って、薬も飲んだけど、全然効かないみたい。今回の風邪はしつこいよ。