Hello student,
You have not mentioned which state you belong to. With a score of 350, you have chances of procuring a seat in the private medical colleges when comparing the results of previous years. But, it depends largely on the state you choose. For example, With this score you have chances of procuring a seat in the colleges of Uttar Pradesh while it is not so in the case of states like TamilNadu for which the scores are extremely high even in private colleges. Also, one thing to keep in mind is that scores and ranks vary each year with difficulty level of the question paper and availability of seats and state quotas. However you can go through the following link to know about the cut off's and ranks for the previous years and calculate your chances of selection
https://medicine.careers360.com/articles/neet-cut-off/amp
Good luck