प्यासी definitely has the same connotations. 'हवस' and 'जिस्म की भूख' are not exactly innuendoes. They straight up mean 'lust' while प्यास is used to 'suggest' it.
In fact, the use of 'thirst' as innuendo in Indian languages goes back to Puranic literature where तृष्णा (the Sanskrit word for thirst) is frequently employed for this purpose.
That said, the word can also be used to refer to chaster desires. For instance, you can find 'दर्शन की प्यासी' (desirous of a vision) in Bhakti (devotional) literature.
Why cant we just say मुझे प्यास लगती है (for the present tense) or मुझे प्यास लगी। (can the है be omitted if we were talking about past tense) or मुझे प्यास लग रही है (if we are talking about the present continuous tense) ?? And which grammatical tense does the sentence "मुझे प्यास लगी है" talks of ?
मुझे प्यास लगी है is the present perfect tense. It is somewhat preferred over the present continuous while talking about the state of being thirsty (similar to 'मैं बैठा हूँ' for I am seated). मुझे प्यास लगती है is the simple present tense meaning 'I (usually) feel thirsty'. 'मुझे प्यास लगी' is a valid past tense form but again, the past perfect form 'मुझे प्यास लगी थी' is preferred to say that you were thirsty in the past.