The adapter that will probably come with that card you shouldn't have a problem using, The 4070 doesn't consume nearly as much power as the 4090, and I personally haven't seen any 4070 or even 4080 run into a melting connectors.
The connector doesn't make a vary solid connection well the older style, but it does make a connection, something that will draw a ton of power like the 4090 could have issues, but the 4070 with the not so solid connection should still be enough of a connection to keep the 4070 happy without any issues.
I wouldn't be worried, the 4070 just don't use a ton of power.
But if you are, time to look for a new PSU, thats really the only option at that point.
Good Luck!