First, if the razor happens to be all metal then you can place the whole thing in a pan of boiling water for 10 minutes to let the dirt and grime loosen up a bit. Then take a brush to it and scrub the heck out of it. The brush you use does not have to be anything special. An old tooth brush will do just fine.

Scrub off all the loosened dirt and then take a cotton swab to get into any engravings or decoration that are holding stubborn dirt. Depending on the way the blade was stored you may have to use some type of degreaser on it to clean it successfully.

If the straight razor is badly rusted you may want to clean it up by using a grinder and taking the rust off that way. you could sharpen the blade at the same time unless ther is still some cleaning to do then wait and sharpen the blade at a later time to save your fingers and hand from cuts while you are handling it.

The goal of historical European martial arts can be understood to be to simplify the infinite complexity and possibilities of actions, into a set of limited, internally consistent rules. I think this is just a necessary step to speed up the decision-making process during combat, to reduce hesitation and allow one to commit faster and more fully to actions. Less thinking, more doing. This is not to say that there is not a ton of thought involved in crafting, and learning the set of rules that prepare you for a single moment in time.