Cheryl, you might try the non-woven, maroon colored sanding pads made by 3M and others that are sold in industrial supply houses, if your home center doesn't carry them. These are less abrasive and tend to ease the transition around the defect areas.
The 320 is OK if you use it lightly. What it means by "sanding between finishes" is between coats within a relatively short time period, like a day or so between when the poly is still soft. Polyurethanes slowly gain hardness and, though it's hard to measure, never really become completely cured.
With furniture, wax is an answer to your problem. You can lightly sand or rub out with the sanding pads and apply a coat of Johnson's wax anytime after about 24-48 hours after the final coat of poly. Don't do this on floors, though, because of the danger of slipping and sliding. And, in my opinion, never, ever use a wax or shining agent, like Future Floor Wax, that contains silicones. These make the floor look great for a short while but make it impossible to refinish.
To sum up, try the pads. I think that they will work well for you. One other suggestion is to walk your dogs often, or trim their nails to prevent or lessen the problem. The walking is better exercise for you than refinishing floors also.