We have purchased the $20/month service during the purchase of our server from Liquidweb. But still after registering from their panel with a new gmail id but unable to get the $20 service in our Cloudflare dashboard, the domain we registered still showing free service.
May you should contact Liquidweb as you purchased the service from them and not directly over CloudFlare. If you bought it from them they should be your first contact point.
But may you want to tell me why you bought a service from Cloudflare from them and not directly over CloudFlare?
Cloudflare has multiple partners that are allowed to resell cloudflare service through their site, they get to have the cloudflare dashboard within their own panel, imho most of the time the implementation is a bit clunky.
The main reason for an upgrade through one of these could be that sometimes you get a free feature such as railgun.
true, Railgun (last time I used it was about 3-4 years ago) is speeding up dynamic conent and is therefore very powerfull when it comes to Shops/Forums. But I to be honest would never go with some third party reseller as you can not rely on Cloudflare support. Even if you ask us here now, we are not common with what Liquidweb looks like and how their dashboard looks like and how to resolve things.
You are on your own. If you are pro and know what you do I would maybe even recommend it but they seem even to struggle to provide the service you bought after you payed them.