To solve this problem using Gurobi, we need to define the variables and constraints according to the given conditions. We'll use Python as our programming language to implement the model.

First, let's import the necessary libraries:

```python
from gurobipy import *
```

Next, we create a new model:

```python
m = Model("Organization_Problem")
```

Now, we define the variables. Since all hours worked must be whole numbers, we'll use integer variables for each person:

```python
Hank = m.addVar(vtype=GRB.INTEGER, name="Hank")
Peggy = m.addVar(vtype=GRB.INTEGER, name="Peggy")
Ringo = m.addVar(vtype=GRB.INTEGER, name="Ringo")
George = m.addVar(vtype=GRB.INTEGER, name="George")
Mary = m.addVar(vtype=GRB.INTEGER, name="Mary")
Dale = m.addVar(vtype=GRB.INTEGER, name="Dale")
```

We'll now add the constraints based on the organization scores and work quality ratings. However, without specific coefficients for each person's contribution to these scores, we cannot directly translate all given conditions into Gurobi code. For demonstration purposes, let's focus on a few key constraints:

1. Organization score constraint (example):
   - Let's say the total combined organization score from hours worked by Hank, Peggy, and George must be at least 57.

2. Work quality rating constraint:
   - The total combined work quality rating from hours worked by Ringo and George must be less than or equal to 368.

Given the lack of specific details on how each person contributes to these scores, we'll assume hypothetical coefficients for illustration:

- Organization score contributions: Hank (1), Peggy (2), George (3)
- Work quality ratings: Ringo (10), George (20)

```python
# Example organization score constraint
m.addConstr(Hank + 2*Peggy + 3*George >= 57, name="Org_Score_Min")

# Example work quality rating constraint
m.addConstr(10*Ringo + 20*George <= 368, name="Work_Quality_Max")
```

To fully solve this problem, you would need to replace these examples with the actual constraints and coefficients based on your specific data. Since we're missing detailed information about how each person's hours worked contribute to the various scores and ratings, let's proceed with setting up a basic structure for solving the organization problem:

```python
# Objective function - For demonstration, minimize total hours worked
m.setObjective(Hank + Peggy + Ringo + George + Mary + Dale, GRB.MINIMIZE)

# More constraints can be added here based on the specific requirements

m.optimize()
```

Please note that without explicit details on the coefficients for each constraint, we cannot provide a complete and accurate model. The following code block is a placeholder based on our discussion:

```python
from gurobipy import *

def solve_organization_problem():
    m = Model("Organization_Problem")

    Hank = m.addVar(vtype=GRB.INTEGER, name="Hank")
    Peggy = m.addVar(vtype=GRB.INTEGER, name="Peggy")
    Ringo = m.addVar(vtype=GRB.INTEGER, name="Ringo")
    George = m.addVar(vtype=GRB.INTEGER, name="George")
    Mary = m.addVar(vtype=GRB.INTEGER, name="Mary")
    Dale = m.addVar(vtype=GRB.INTEGER, name="Dale")

    # Add your constraints here
    # Example constraint
    m.addConstr(Hank + 2*Peggy + 3*George >= 57, name="Org_Score_Min")
    m.addConstr(10*Ringo + 20*George <= 368, name="Work_Quality_Max")

    # Objective function - For demonstration, minimize total hours worked
    m.setObjective(Hank + Peggy + Ringo + George + Mary + Dale, GRB.MINIMIZE)

    m.optimize()

solve_organization_problem()
```