Can i sign into my old 360 account on xbox one with only the old gamertag?

Each 360 account can be associated with a Live account.

Currently I have:

old 360 account -> associated with my Live account but with the old gamertag

new 360 account -> associated with my Live account but with the new gamertag

The new account got created when I downloaded my profile using my microsoft sign in. The old account was associated with the same microsoft sign in but it didn't recognize my gamertag change. Now it won't let me sign in to live because it asks for my microsoft
sign in details. I'm guessing the sign in fails because it's trying to find the microsoft account associated with the old gamertag.

I either need to somehow transfer my saves over to the new 360 account so I can upload them to the cloud, or I need someway for my Xbox to realize that my gamertag changed on the old account so I can sign into live again with that account.

No. You would need the email and password. You can try going through here - https://support.xbox.com/en-US/my-account/microsoft-account/forgot-microsoft-account-solution
to see if you can find the email and then reset the password.

Another thing you can do is contact xbox support by going to support.xbox.com/contact-us and give them your gamertag and if they can verify the account they can give you the email address associated with the account.

If you follow the instructions on here
support.xbox.com/.../sign-in-to-another-console then you can log into your old xbox 360 account on your xbox one. As for playing the old xbox 360 games, check this site for the full list of xbox 360 games that will play on an xbox one

www.xbox.com/.../backward-compatibility

On my new xbox I've redeemed my old account but I've changed the gamertag, I'm wanting to transfer games ext from my old xbox to the new one but the data is owned by my old gamertag can I still transfer from the old gamertag to my new one ??