Should I insulate the floor of the attic of the sloped sides of the roof? I also have dampness issues. The house is under insulated and is also drafty.

A: It sounds like you have two different problems: insulation and moisture. You need to deal with both. My fundamental question is where is the moisture coming from? The usual culprit is the basement. If the basement is damp (in summer when the humidity is high dampness in the basement is usually caused by warm moist air condensing on the cooler surfaces (especially cold-water pipes) and in the winter it is caused by water wicking up from the ground. This, in turn is often caused by leaking guttering and downspouts that dump the water right by the basement wall. Leaking pipes cause dampness year round. 
First, fix the source of the dampness. So get downspout extenders and clean out the clogged leaves in the gutters and maybe fit perforated plates on top of the gutter, and fill any cracks in the basement wall with exterior grade spray foam. I did this on my rental property and it greatly helped. Then I would seriously consider replacing your hot water tank (I am assuming it is heated from your furnace) with a heat pump hot-water tank. A Rheem 55 gallon hybrid heat pump hot water tank at Home Depot costs about $1200, you can get a $600 rebate on this. It will take an hour or two with an electrician and plumber to install it. These heat your water at less than the cost of heating oil and about the same as heating with natural gas. However they also dehumidify your basement and that is the big plus. Ideally you would also insulate the ceiling of your basement (just push fiberglass in between the rafters or ask MassSaver to do it for you) to prevent the heat from your house just leaking down into the basement and becoming the source of heat for your heat pump hot water tank. This is a good fix for you but if you also get solar panels you would be saving a ton of money on heating your hot water, even if you are on natural gas today. 
With the moisture dealt with, I think it would be fine to insulate the attic  either on the floor or the sloped sides of the roof. Which you do depends mostly on whether you use the attic as storage space. If you do, the sloped side is better. If ever you intend in the future to install a heat pump for heating the house with an air-handler unit in the attic you MUST insulate the sloped part of the roof or you will get icicles and ice dams in winter. You could either wait a season to see if the moisture fixes work or insulate the slopes with fiberglass or Rockwool because they are permeable to air and so will allow a small amount of condensation to evaporate. The best solution is to add a one-way vapor permeable membrane (but 100% air tight) on the inside of the rafters/insulation such as Intello which you can get from Building 475 in Connecticut. You need a pro to install this and I have consulted with Dolphin but I have not hired them yet because I need to deal with the water issues in my basement first! This membrane prevents moisture traveling from the loft to the roof surface but still allows any moisture from behind the membrane to evaporate to the inside. This insulation-plus-Intello-approach is about the gold standard in roof design and I am about to use it on my other rental property. This is going to be more expensive than just blowing in cellulose to the floor of the attic, but it will still save you loads of money on the bills and give you a completely air tight roof and get rid of the condensation problems in the attic. However, still fix the source of moisture first, because that moisture is causing mold elsewhere in the house, condensation on windows and probably even making the towels stay damp on the rack too.