I like KMC Z chains. They are inexpensive and well made. An 8 speed chain (made for bikes that have 8 gears in the rear) works well on bikes that have less than 8 gears in the rear, like 5, 6, and 7 speed freewheels. If you go looking for a 10 speed chain, you'll likely get one that is made for bikes that have 10 gears in the rear. This chain would be too narrow for your purposes.
Also, typically the chain and freewheel should be changed together, as they both wear and a new chain on a worn freewheel can produce problems. The issue you describe may or may not be related to the condition of your chain. It's best to have a mechanic check it out and see what's what if you aren't sure.
Best of luck.