301 redirect via htaccess rules on the new site

What you want to do is Domain Name Forwarding, and it CAN NOT be done using .htaccess of the target domain. If you are using godaddy to register your domain, here is a guide that they have prepared for people who would want to forward one domain to another:

https://ph.godaddy.com/help/manually-forwarding-or-masking-your-domain-name-422

The process is about the same for other domain name registrars. You have 2 options when doing the domain forwarding. One is to keep showing the original domain, such as abc.com in your example. This is Forwarding with Masking. The other option is to forward abc.com to abc.org and the user sees abc.org in the browser. This is Forwarding without Masking.