Get date if category and month match

I have a workbook with Checking account register on sheet1, Budget on sheet2. Checking columns are Date, To/From, Category, Credit, Debit, Balance. Budget columns are Due Date, Category, Amt Due, Paid. I am trying to pull the date from checking on sheet1 and insert it in the budget column paid on sheet2, provided the category and month match on sheet1 and sheet2. Hope that made sense. Attached is the excel file. Really appreciate any and all help.

One can adjust it to the proper columns.

Sub post()

Dim ck, bg As Worksheet
Set ck = Sheets("Checking")
Set bg = Sheets("Budget")

r = 3
While Cells(r, 1) <> ""
If ck.Cells(r, 8) = "" Then
m = Month(ck.Cells(r, 1))
ck.Cells(r, 9) = m
ct = ck.Cells(r, 3)
With bg.Range("b:b")
Set c = .Find(ct, LookIn:=xlValues)

If Not c Is Nothing Then
fa = c.Address
Do
dt = (bg.Cells(c.Row, 1))
dm = Month(dt)
r2 = c.Row
Set c = .FindNext(c)
Loop While dm <> m And fa <> c.Address
If dm = m Then
bg.Cells(r2, 8) = "Paid " & Month(dt) & "/" & Day(dt)
ck.Cells(r, 8) = "**"
End If

End If
End With
End If
r = r + 1
Wend

End Sub

Thank you rcm and Jeff. I removed the formula from the paid column on the budget sheet and tried the code. Here is a screenshot. Added info to column H but it matches the due date, not the date paid. Let me know if there is something else I should try with the code. Thanks again. NOTE: info in column F is what I put there for reference and is the paid date from checking (no code involved).
Capture.PNG

revisions: if there is already a posting for the given category it searches for the next one. If cannot find one not having a post, it sticks the payment to the last one in the category. The accumulated expenses was added as part of the testing.

Sub post()

Dim ck, bg As Worksheet
Set ck = Sheets("Checking")
Set bg = Sheets("Budget")

r = 3
While Cells(r, 3) <> ""
  If ck.Cells(r, 8) = "" Then
    m = Month(ck.Cells(r, 1))
    d = Day(ck.Cells(r, 1))
'    ck.Cells(r, 9) = m
    ct = ck.Cells(r, 3)
    With bg.Range("b:b")
      Set c = .Find(ct, LookIn:=xlValues)
      If Not c Is Nothing Then
        fa = c.Address
        r2 = c.Row
        Do
          dt = (bg.Cells(c.Row, 1))
          dm = Month(dt)
          r3 = r2
          r2 = c.Row
          Set c = .FindNext(c)
        Loop While (bg.Cells(r2, 8) <> "" Or dm <> m) And fa <> c.Address
        If dm <> m Then
          r2 = r3
        End If
          bg.Cells(r2, 8) = bg.Cells(r2, 8) & ", " & m & "/" & d
          bg.Cells(r2, 9) = bg.Cells(r2, 9) + ck.Cells(r, 5)
          ck.Cells(r, 8) = "**"
        
      End If
    End With
  End If
  r = r + 1
Wend

End Sub

Thank you rcm. I sincerely appreciate your time and effort. I deleted the formulas in the budget sheet paid column and ran the new code you provided. Here is a screenshot. Results of the code are in columns H and I. It does not seem to find all the categories and dates. If there is anything I need to do please let me know. Is there a way to put the results in the paid column? Thanks again.
Capture2.JPG

"The non budgeted items could be added to list flagging them as "NOT budgeted". Hence, you could have sum the budgeted totals and the expensed totals in the same page." Forgive me but I am not sure I follow you on this.
I added some budget items and flagged them in checking but the paid and amount results did not show even after I ran the macro (module). Is there a work around to flagging them in checking column H? Maybe a list of budgeted items that could be named? Hope that made sense. I really appreciate your help.
I think automatic routine would be best.

You are amazing, rcm. Thank you.
If checking column H is ** the item is included in budget, if it is xx it is not, correct?
The dates in budget paid column are repeating the same date(s), see screenshots.
I entered a groceries transaction in checking and it showed in budget but when I deleted it from checking, as if it was already entered or a mistake, it did not delete from the budget worksheet. Changing the category on the budget worksheet does not change the data, even after running the macro. Adding a category to the budget sheet and changing the transaction in checking from xx to ** had no affect. I do not mean to be overly picky or seem unappreciative, rcm, because I do apprecite all your time and effort. Just wanted to put the workbook through some normal use.
Capture3.JPG
Capture4.JPG

1. The xx and the ** are posting flags done by the routine to get a complete run one must CLEAR column H.
2. The redundant dates are the result of me not clearing the column D in budget before each run. Sorry.
3. Now, the tricky part: updates can only be done through a complete rerun (with the code posted). Updating code could be written though. But it is far more easy to write a cleaning code (to clear the H column in the checking tab and the dates and values in the budget tab)

4. I improve the routine with clearing all flags and dates and running the whole works. To make it run automatically change any value in the checking tab. If you want to change category do so but rewrite the value (thats the trigger)

Wow. Thank you, rcm.
I added a category to budget and did a few transactions in checking, manually ran the macro and all seems well. I will test more later, Sunday night is family time. I had to delete a row from budget when I input the wrong date and it messed up the groceries category and result when I did (sorry). See groceries dates 3-13 and 3-28. Would there be any way to pull this data into a monthly report? Could there be an automated way to update the data? Not sure if family members will remember to run a macro (they don't know what a macro is). I apologize if I am asking too much. If I am, please let me know.
Capture5.JPG

I did some quick reporting but if it is not what you want please feel free to send over the expected output layout at your leisure.

One thing, the copy I sent runs the macro if the debit or credit values are changed or retyped...

Hello RCM, Apologies for the delay. Very hectic at work. The latest rev looks great, thank you. I have some reporting ideas that are on the excel workbook I actually use. I did not post it here for privacy reasons. I will add them to the workbook we are working on and upload it. Probably will not be until this weekend. Thanks again for all your help.

You are welcome. Keep in touch if you need more help!!!

Appreciate your patience, RCM. And all your help. Here is the latest from me with the budget report I am using on another worksheet. I added a sheet called Notes to the workbook where we can put ideas, questions, comments, etc.