Is there any way to get my old games onto a new account without re-buying these games?

Unfortunately without access to that account, it would not be possible to get those games.

Nevermind! Once I downloaded my old profile onto the new xbox, I was able to re-download the game onto the harddrive

I bought it off of Minecraft’s site through my microsoft account, so if I happened to buy it on the wrong account, is there any way to refund the money or get the game onto my account? I have looked into my different microsoft accounts but none of them have the game that I purchased.

What im saying is if i do this i can Re download DLC which i brought from my old account onto my new account ?

The only way to use the games on the new console would be to "find" access again to your old account and transfer the licenses to the new console. After which you would be able to play the games on your new tag from your new console. But if you have no way
of getting your old account back, There's not a lot you can do besides buying the content again on your new account.

If you can give some information about your lost account, As in why/how you lost it. Whether just an email issue or a ban, There may be a way for you to gain access to it again in order to do this. But if it is a friends/ex partners/ or family members account
i wouldn't suggest trying to get onto it.

Apart from that, it is possible to change the emails associated with accounts, and other measures which may help If we know what the issue is. Good luck with your gaming!