Forgotten Realms date calculator

Python 3, 712 652 636 567 563 552 550 548 529 540 bytes

At last, I found time to write an answer for this excellent question. It isn't very golfed yet (the month names list and the number of days list are is particularly egregious in this case, and the fact that handling negative D requires a separate while loop) but at least it's an answer.

Edit: Fixing a bug

def h(s,n):
 x=s[:-3].split(", ");x=[1]*(len(x)<3)+x;t="Deepwinter,Midwinter,The Claw of Winter,The Claw of the Sunsets,The Claw of the Storms,Greengrass,The Melting,The Time of Flowers,Summertide,Midsummer,Shieldmeet,Highsun,The Fading,Highharvestide,Leaffall,The Rotting,The Feast of the Moon,The Drawing Down".split(",");p,q,r=int(x[0])+n,t.index(x[1]),int(x[2]);d=[30,1,30,30]*4+[1,30];d[10]=r%4<1
 while p>d[q]or p<1:a=[-1,1][p<1];q=(q-a)%18;p+=a*d[q-(a<0)];r-=a*0**q;d[10]=r%4<1
 return', '.join([str(p)]*(d[q]>2)+[t[q],str(r)])+" DR"

Ungolfed:

def harptos(date, num):
    t = "Deepwinter,Midwinter,The Claw of Winter,The Claw of the Sunsets,The Claw of the Storms,Greengrass,The Melting,The Time of Flowers,Summertide,Midsummer,Shieldmeet,Highsun,The Fading,Highharvestide,Leaffall,The Rotting,The Feast of the Moon,The Drawing Down"
    t = t.split(",")        # split up the names of the months
    x = date[:-3]           # removes " DR"
    x = x.split(", ")
    if len(x) < 3:
        x = [1] + x         # if we have two items (holiday), append a "day of the month"
    p = int(x[0]) + num     # initialize the "date" by adding num to it
    q = t.index(x[1])
    r = int(x[2])
    d=[30,1,30,30]*4+[1,30] # all the month lengths
    d[10] = r%4 < 1         # leap year toggle
    while p > d[q]:         # while the "date" > the number of days in the current month
        p -= d[q]           # decrement by number of days in current month
        q = (q+1)%18        # increment month
        r += 0**q           # increment year if the incremented month == the first month
        d[10] = r%4 < 1     # leap year toggle
    while p < 1:            # while the "date" is negative
        q = (q-1)%18        # decrement month first
        p += d[q]           # add the number of days in the decremented month
        r -= 0**q            # decrement year if the decremented month == the first month
        d[10] = r%4 < 1     # leap year toggle
    m = [t[q],str(r)]       # start the result array
    if d[q] > 2:
        m = [str(p)] + m    # if the month is NOT a holiday, add the day
    return ", ".join(m) + " DR"

Ruby, 543 523 521 498 511 509 bytes

To encourage more answers to this question, I'm going to post a Ruby version of my Python answer, since I figured it would be shorter. This answer is shorter but not by much. Can you do better?

Edit: With thanks to Martin Büttner and his suggestion here.

Edit: I golfed the "number of days in a month" list down considerably.

Edit: While golfing down how I handled d[10]=r%4<1?1:0 to d[10]=0**(r%4) for a byte, I noticed I'd introduced a bug while golfing down d, the number of days list, so that Shieldmeet had 30 days by accident. And so, the byte count has come back up. I will also edit the Python answer to fix this bug there.

Edit: I forgot that functions don't need to be named in this question.

->s,n{x=s[0..-4].split(", ");x=x[2]?x:[1,*x];t=(["Deepwinter,Midwinter","Winter","Sunsets","the Storms,Greengrass,The Melting,The Time of Flowers,Summertide,Midsummer,Shieldmeet,Highsun,The Fading,Highharvestide,Leaffall,The Rotting,The Feast of the Moon,The Drawing Down"]*',The Claw of ').split(?,);p,q,r=x[0].to_i+n,t.index(x[1]),x[2].to_i;d=[30,1,30,30]*4+[1,30];d[10]=0**(r%4);(a=p<1?1:-1;q=(q-a)%18;p+=a*d[a<0?q-1:q];r-=a*0**q;d[10]=0**(r%4))until(1..d[q])===p;z=d[q]<2?[t[q],r]:[p,t[q],r];z*", "+" DR"}

Ungolfed:

def h(s,n)
  x=s[0..-4].split(", ")
  x=x[2]?x:[1,*x]
  t=["Deepwinter,Midwinter","Winter","Sunsets","the Storms,Greengrass,The Melting,The Time of Flowers,Summertide,Midsummer,Shieldmeet,Highsun,The Fading,Highharvestide,Leaffall,The Rotting,The Feast of the Moon,The Drawing Down"]
  t=t*',The Claw of '           # turns the above array into a string with "Claw"s inserted
  t=t.split(?,)                 # then splits that string back up again by ","
  p=x[0].to_i+n
  q=t.index(x[1])
  r=x[2].to_i
  d=[30,1,30,30]*4+[1,30]
  d[10]=0**(r%4)
  until(1..d[q])===p
    a=p<1?1:-1
    q=(q-a)%18
    p+=a*d[a<0?q-1:q]
    r-=a*0**q
    d[10]=0**(r%4)
  end
  z=d[q]<2?[t[q],r]:[p,t[q],r]  # putting z=[t[q],r] on another line saved me no bytes
  z*", "+" DR"
end