I'd be happy if someone would answer my question. Thanks in advance.

  Are these two sentences the same in meaning?

A:  I accept your offer.

B:  I would accept your offer. --- Second one is hypothetical. It means the offer has not been accepted yet.

  Again, are these two sentences the same in meaning?

C:  I apologize.

D:  I would apologize. --- Same reason with the above.

  Yet again, are these two sentences the same in meaning?

E:  It's going to rain. I advise you to take an umbrella with you.

F:  It's going to rain. I would advise you to take an umbrella with you. --seems meaningless to me in this sentence.

   I think A, C and E express actions (accepting /  apologizing / advising) which end immediately after saying. --- In a dialogue you can tell all of them without any problem.

   But how about B, D and F? Do these have the same function as A, C and E? -- no for example if  you say I would accept your offer that means in real time you haven't accepted the offer, but if you were given the chance, you would accept.
   (Is A the same as B?   / Is C the same as D?   / Is E the same as F?)

   I think B, D and F just help say in a roundabout way. --- Rather than a roundabout way, I see them as a supposition of an unreal situation.        

This is my try.