Hello Aspirant,

Aakash Institute

Allen Career Institute

Resonance

PACE IIT and Medical

Career Point

Narayana IIT Academy

These are few best coaching centers for NEET preparation in India.They several  have branches all over India. You may visit their official websites and get the details about the branch you wish to join according to your convenience.Weekend or weekdays class schedules should be chosen according to what suits you , if you contact them they will assist you in deciding what best suits your requirements by determining the level of preparation needed .Apart from these , you may search for centers in and around your locality and if you are forced to go far away for coaching , then you can maybe stay with your relative or a family friend's house for the time period or opt for online classes . If you are dedicated and disciplined in managing time then you can make online coaching work. Now to the second part of the question, to balance both board and NEET requires again, discipline, dedication ,prioritization and time management. You should for sure focus more on your boards and if you are thorough with your board syllabus ,it will cover 50-60 % of you NEET preparation.  Solve as many sample papers and previous year question papers as possible.Divide and use your time judiciously and most importantly avoid unwanted distractions . But do not forget to take a good amount of short productive breaks (like sleep, exercise,doing house chores,spending time with parents or talking to friends etc) while studying , do not overload your brain.Of course you can watch movie or listen to songs , but have a control on that.Do not compromise on food and sleep whatsoever. Avoid thinking about exams or NEET when possible ,mainly while taking your break and most importantly do not consider clearing NEET ,your ultimate goal, because that will create over expectation and anxiety.

https://medicine.careers360.com/articles/neet-preparation-tips

https://medicine.careers360.com/articles/how-to-prepare-for-neet

You may checkout the link to NEET coaching by Carers360:

https://learn.careers360.com/neet-coaching/

All the best