Ok, take a deep breath. All is not lost.
Anchor Worms typically do not float visibly around in the water - even when they "come in on the plants". An adult worm typically is already attached. The ones that travel are the teenyweeny juvenile ones...
What you may be seeing is planeria. These are tiny little white worms that are often found in tanks. They clean up detrius and waste in a tank that has lots. They are harmless to the fish - in fact harmless to everything except to the esthetics of the tank.
A fish in a bowl (good for you on getting the 10 gallon tank!

) will need LOADS of water changes to keep his water in good shape. So, you do not have to be afraid of changing out the water. I would suggest that you take a syphon tube and carefully clean the gravel that holds your plants in the bowl and replace the old dirty water with new water. This will refresh the bowl as well as remove some of the detrius that the worms are feeding on.
Keep an eye on your fish. When a fish is in a very SMALL volume of water, such as a bowl, not only do the water parameters need to be monitored on a DAILY basis, you will find that any parasites that may be around will attack with a vengence.
If you see a white, stringy looking thing hanging from your fish - with the head of it buried under a scale - usually making a reddish looking sore there, THEN you have anchor worms.
Anchor Worms are relatively easy to erradicate - so do not panic.
Ok?