Can I transfer my old minecraft world from my old account to my new account on PS4

There is no official way to do this. Your worlds are tied to the save data of the account they were created on. Either use your old account or make a new world. Sorry that this doesn't much help, but there really isn't much you can do.

If you have Minecraft Bedrock for Android, iOS, or Win10 and now that PS4 finally has realms support, there is a way you could do this (provided the world in question doesn't use Minecraft marketplace content). It's a slightly pricey option as it require purchasing two realms (one on each Microsoft account tied to each PSN account) and that one PSN account primary has PSPlus status. You can cancel the realms after you've transferred your saves.

On your PS4's old PSN account, link it to a Microsoft account (NOT your primary Microsoft account, because this is your old account you don't want to use any more and Microsoft accounts can only ever be linked to PSN accounts one time), buy a realm on that Microsoft account and upload your world to the realm.

Now login to the same Microsoft account tied to that realm on your computer or phone, and download the world onto your computer or phone.

On your new PS4 account, link it to your primary Microsoft account and buy a realm on that account.

Login with that Microsoft account on the Win10/Android/iOS device where you downloaded your save and replace the world on the realm you just bought with it.

You should now be able to download the save from the realm on your new PS4 account.

Repeat the process for each save you want to transfer.