If you attempted this before, then I think you didn't understand exactly what was happening.
If you change the nameservers to dreamhost's, then you will make all DNS edits via the dreamhost panel, you won't be able to edit DNS via godaddy.
Who the domain's registrar is, and whose nameservers the domain is set to really don't have anything to do with each other. If the domain registrar is godaddy, that's who you pay your annual fee to, and that is where you log in to change domain whois and privacy info as well as where you set the nameservers.
Whose nameservers are in play determines which is managing the individual DNS records (A, AAAA, CNAME, MX, TXT, etc). If you have GD's nameservers then you would use GD's tools to make changes to specific DNS records, if you have dreamhost's nameservers then you will use dreamhost's tools to make changes to specific DNS records.
Regardless of which, both hosts have the ability to manage some records for you when you are using their DNS. Because GD's nameservers are your current choice, then the records that dreamhost automatically generates for you have no effect. You have to copy them over, as you have done using the screenshot you provided earlier. If on the other hand, dreamhost nameservers were in play then those records would automatically be included for you, and you would only need one more A record, to point the domain to your home server. Since DNS records for mail can get complicated, using dreamhosts DNS would mean that you don't have to manage those records yourself, just that single A record to point back to your home server.
AFAIK.... the only drawback to using dreamhost for DNS, instead of GD's as you have it set now, is that dreamhost doesn't let you change the TTL value on your own, you have to live with the dreamhost default which is 14400 or 4 hours. This is really only a problem during the 4 hours following a change (and you don't change things very often). People get excited because they can make the value less, but it really isn't something most people need.