You can keep the old domain but if you have the same content on both domains, you will not get good rankings because of the duplicate content. You can always do the 301 redirect to the new domain so that you will not lose visitors and all the backlinks and SEO that you have worked on will be preserved. Here is the guide on how to do this: creare.co.uk/blog/seo-advice/how-to-301-to-a-new-domain.
I have never used the paid Alexa but I can see that their pricings are set really high so very few affiliates can afford them and especially the newer ones. They have a free fourteen days trial so you can test them and if you think that their software gives you a good value for the money you pay them, you might keep using them but there are a few free tools on the web, that have limitations but they can give you some data that you can use.