I need new psn account and subscribe PS+. Will I be able ot access the games from my old PS+ account?

As long as an account is subscibed to PS+ it has access to offered games. Having PS+ on account A does not grant access to PS+ games of account B.

If you keep PS+ on your original account and make your console primary for it, you will be able to access downloaded games and have multiplayer capability unlocked for every other account on that console.

So as long as your original account has active PS+ you will have access to previous games (can be downloaded via that account, then accessed with any other on that console). If PS+ expires, access to the account's games will be lost, regardless of PS+ status of other accounts.

"Family member" functionality is used for parental controls and does not grant any additional game sharing options.