2001 Ford Taurus Car is Shaking

Hi,

My car does not start to shake until I reach about 70 on the freeway. However the shaking stops after a few minutes, I thought that it could have been just my tires sitting in the cold weather and getting flat spots and the shaking is from the flat spots rounding out. But this is a reoccuring problem. I recently went to get my oil changed and the service shop told me that my battery voltage was very weak. I would like to know whether my battery being weak may cause my car to shake which in this case I could fix it by getting a new battery, or if there is another problem. I am leaving for vacation soon and I dont want my car to die on the trip.

Thank you for your time and advice
Sincerely,
Josh