Rear Basket Questions

I traded in the old lady shopping cart for the Wald 535 Rear Basket to get groceries... Great idea. Problem is I lack wrench turning skills and often have brilliant (aka terrible) ideas. While installing this beast of a basket, I thought that I could attach the legs to both the rear wheel stud & to the frame. Nope! Duh! So, I put the wheel back in its place and finally got the rack attached.

Went to the grocery store couple days later and the bike 'wobbled' like none other, with a u-lock in each basket. It was slightly off balanced load but less than 8# total. Got a full load of groceries and wobbled on home. (I was on the sidewalk and at a slug-pace set by the kid.) Tried again last night (without the kid) and got on the road. With empty baskets, I still shook and it increased with my speed. (Rode a block with 34# of dog food because I am stubborn & too lazy to carry, see pic.)

So my questions:

• Is the 7# basket too much for my 1980s Fuji (steel) Sagres? (If so, is it the bike, too skinny tires??)
• Am I just a dumbass that doesn't know how to put a wheel on straight? (Gotta a video, link that could help this idiot out for proper installtion? My commuter is a single speed-all these gears must have thrown me off!?)
• Do I need a Stabilizer?!?
• What else could I have done wrong?

Before I give up and pay for someone to look at it, I will take off the basket see if that is the problem. I was just looking to see what else it could be.

Thanks! Jenn