"I think that dog is sick."
When you want to say "I think..." in Japanese, you use ~と思う/と思います
For verbs and i-adjectives, you just use the plain forms (you can change the tenses, but you don't use the formal forms): 明日（あした）は雨（あめ）が降る（ふる）と思う（おもう） I think it will rain tomorrow.
サッカーするのは楽しい（たのしい）と思う（おもう） I think playing football is fun.
But for nouns and na-adjectives, you need to add a だ (or だった for past tense). This is also always the plain form:
あの人（ひと）は学生（がくせい）だと思う（おもう） I think that person is a student.
あの女（おんな）の人（ひと）は綺麗（きれい）だと思う（おもう） I think that women is beautiful.
You only change 思う/思います to reflect formality in all of these cases
I am not certain but 病気 is a noun you can turn into a verb by adding suru. But verbs have to usually be at the end of a sentence and in this case it isn't. When you want to use a verb and it isn't at the end of the sentence you should use plain form conjugations. Example 食べる instead of 食べます。 For non verb that you wish to come at the end of a sentence you often use です proceding the non verb. And when you want to put a non verb some place other than the end of a sentence and say that non verb is something you use だ which is the plain form of です。
Again not certain of what I just wrote but that how I currently understand it.