Learn Russian: Мне пришлось искать другой путь.

путь is a special anomalous noun. Even though it ends in a soft sign (-ь) and declines like a feminine noun in most other cases, its gender is masculine. The adjective другой matches this by taking the standard masculine form.

What case is другой путь in this sentence?

It is in the accusative case because it is the direct object of the verb искать (to look for). Since путь is an inanimate masculine noun, its accusative form looks exactly like its nominative (dictionary) form, and so does the adjective другой.

Why does the sentence start with the dative Мне instead of Я?

To say "had to" in Russian, you use the impersonal verb прийтись (here in the neuter past tense: пришлось). The person who is obligated to do the action must be in the dative case, so Я becomes Мне. Literally, it is like saying "It fell to me to look for..."

Could I use the perfective verb найти here instead of искать?

It would change the meaning. искать is imperfective and focuses on the process of "searching" or "looking for". If you used the perfective verb найти, the sentence would mean "I had to find another way" rather than just looking for one.