I think this is one of the hurdles we all have to jump when learning guitar - or even when you can already play some but are just learning a new chord shape.
The G chord was a tricky one for me so I cheated.
I play G with 1st finger on the high E or sometimes both E and B, at the 3rd fret.
Then use thumb to fret the E string at 3rd fret, and damp the A with it too.
When I was starting out (yeeeears ago) this got me though that "having to slop down" thing. It also allows me to move gtom G to D quite quickly because the relevant fingers are in the right area aldreay. I've later found that it also leaves a couple of finger free for playing riffs etc.
I can play a "normal" G (a string at 2nd fret) now because I've practiced but frankly I hardly ever do. Purists will probably throw their hands up in horror at this but I say : If we all obeyed the (precieved) rules, we wouldn't have Hendrix.
There are other ways to play chords - eg C : traditionally with open G string (ooer missus) or you can barre it by moving the A shape up 3 frets.
I'd suggest experimenting with chord shapes and find the best that suit you to gain the quickest change- but view it as a way to get things changing quickly, not necessarily as the "Final" way you're going to do things. It;ll enable you to get a flow going and that's where things get really fun :-D