Maybe placing several sheets of newspaper down and covering with mulch would help or leave them and use a weedeater to cut them down before the weeds go to seed?

Good suggestion to use a weedeater to cut weeds down before covering with mulch. Use a scissor if you don't have a weedeater. Mulch is my favorite method of weed control. Once you have it in place it will simplify the job.

You can also try using a hoe to remove the weeds on the side gently & replace the soil there that moves. After you get the current weeds under control cultivate lightly every few days and you won't have large weed rootballs to dislodge the sides.

If you're having a lot of erosion on the sides of the raised beds then the angle might be off a bit. Mulch will help it stay together.

A bale or two of straw would help solve the problems. If you have wood chips available use those. I have filled the paths & sides up to level ground so raised beds don't erode away on a sloped garden. It seems like a funny idea, but really works. Plus you have cushioned paths for knees when working in the garden & you stay cleaner.

Are you watering with a hose? Lightly sprinkling rather than gushing water helps your loose soil stay put. You can bury something next to the your larger plants to fill with water & let seep out to water the bed. I've used 2 liter bottles with bottoms removed then bottom up buried in bed, or 2 liters with small holes in bottom & buried to necks. Empty milk jugs, bottomless coffee cans, empty black 1 gallon pots, even clay pots work. Direct the flow of water into the container still being careful not to wash the soil away.

Another idea is to plant fast growing arugula or rocket to help hold the soil together. It goes to seed & the flowers are pretty. You can leave it be & let it reseed or snip & eat or let fall to be mulch. If you refrigerate the dry seed packet then plant it another day it might germinate better in warm summer weather since it's a cool weather green.