Elizabeth, whether you are finishing a piece of furniture or a floor, the process is the same. It's sand, then finish. Think of your floor as a piece of furniture and follow the same steps as you would finishing or refinishing a piece of furniture. Watch Norm or the Furniture Guys on TV and that will show you the process, then change the materials to what you are working on.
The first thing you do is build the piece or repair or replace any parts that need it. Then you sand to get out any tool marks, mismatches or to roughen the present finish so the new finish will stick. Then you finish and it makes no difference in the process if you are going to paint or stain. If you don't have a good base to work on, you will not end up with a good finished product.
If you want to remove the paint drippings as a temporary measure, then you can use Citristrip or, if the floor is very shiny, you might even be able to scrape it off with an Exacto knife and spot sand. But remember that you will be taking off all the finish in the areas that you worked in and those spots will still show and really should have a finish put back over them.
One of the problems with an on-line forum like this is that you cannot see the problem. It is a heck of a lot easier to advise if you can see if what you say is "a lot of paint drippings" is what I think "a lot" is. That's why I think you should call one or more pro's in to take a look and give you suggestions on how to proceed. As far as whether refinishing your floor is a DIY or a job for a pro depends on your level of expertise, desire and enthusiasm. In any DIY project where you have to make this decision, you are trading off time against money. That's another reason why you should check with the pro's to see what to money factor really is. Then make your decision from there. As an example, at my daughters house, when one pipe froze and broke, I replaced it. I work cheap! :o) When it was time to replace all the piping, she called a pro. He did it quicker and probably cheaper since he had all the tools and equipment with him and knew exactly what he needed before starting the job. It takes at least an hour for a run to the hardware store or home center for Papa and I usually carry three 5 gallon buckets of tools every time I go out there. The pro has a whole truck.
I don't mean to deter you. There is a new "sanding system" of tools available for rent that are much easiers and safer for DIYers available. They probably work a little slower than those a pro would use. Are they available in your area? At what cost? You would want to research that to make a good decision also. Hope I haven't confused you but it sounds like you and your husband are both right. Check out having the pro come to replace the bad boards, then you doing the sanding and finishing. Henry