Question for internet techies

My current business site is with Hostgator. I bought from them both the domain name and the server space. Recently I've been developing a new site on a subdomain of my current domain (i.e. newsite.mysite.com) until it's ready to go live. Now the site is ready, and I switched to a new superfast Cloudways/Vultr host. I have migrated the new site files across to the Cloudways server and I just need to point my existing domain name to it. 
 
So I want to cancel my server hosting with Hostgator, but keep the domain name with them (they sell both server space and domain names, like GoDaddy). So off I went to my Hostgator account to see about pointing the domain name to the new server. I was told all I have to do is create an A record with the domain name and IP of the new server. Quickly got confused in Hostgator's cPanel but did manage to create an A record. 
 
Now I know these things take a while to propagate, but when I check my domain in a DNS propagation checker online, for most locations it's showing BOTH the old Hostgator IP and the new Cloudways IP. Is this normal? I'm worried that I have to manually delete the old records for Hostgator or something. I just spent 45 minutes on a live support chat feed with a Hostgator doofus, and the guy is just being obtuse and confusing and talks complete gobbledegook. 

I bet Cloudways would be more helpful ... they might even do it for you.

That was the worst support fiasco I have ever been through. In the end I opted to ditch Hostgator altogether and move the domain name to Namecheap - they were even trying to suggest to me that if I wanted to move my site to Cloudways there would be problems and I wouldn't have full control over my domain. I said screw that, I'm done with you guys. Now transferring the domain from Hostgator to Namecheap has brought its own set of problems - at one point I had two Live chat windows open for both of them, both beeping with different tones and one support guy asking what the other one was saying, all the while me having to research stuff online. I had about 40 tabs open at one point. 

Yes finally my new site is up and running, with a new domain host and on a new super fast server. I don't know if the DNS has fully propagated yet but at least it seems to be working here in NYC, where my customer base is. You would not believe how bad Hostgator was in letting me transfer the hosting and domain away from them. At one point, the new registrar Namecheap was telling me they couldn't send me an approval email for the transfer because my contact info in Whois was showing the default Hostgator contact info instead of mine. I did a Whois search myself and it came up with all my contact info, exactly as I'd entered it years ago, and I had checked it a few hours earlier whilst researching how to switch hosts - it was all correct. But the new support lady said no, we're using a more up to date Whois report that's updated in real time. So I checked my domain contact info in the my Hostgator settings, and sure enough, Hostgator had apparently changed it at some point during the support call. It's almost as if they deliberately sabotaged it so that I couldn't get approval to switch domains. Oh and they also managed to mess up my MX records which route my webmail through Gmail, so I couldn't receive important emails related to the transfer.
 
Never, ever use Hostgator. A few years ago I logged into my server space and found a folder called something like "Russian family incest." Inside were a bunch of HTML files - no photos - that looked like they were the framework of a porn website. I called Hostgator immediately and they said "we have no idea how they got there - just delete them, should be fine."