## Step 1: Define the symbolic variables
Let's define the symbolic variables for the problem:
- $x_1$ represents the number of digital watches
- $x_2$ represents the number of analog watches

## Step 2: Formulate the objective function
The profit per digital watch is $15, and the profit per analog watch is $10. The objective function to maximize profit is:
\[ \text{Maximize:} \quad 15x_1 + 10x_2 \]

## 3: Define the constraints
Based on the problem description, we have the following constraints:
1. Demand for digital watches: $x_1 \geq 150$
2. Demand for analog watches: $x_2 \geq 120$
3. Production limit for digital watches: $x_1 \leq 200$
4. Production limit for analog watches: $x_2 \leq 180$
5. Minimum total watches to ship: $x_1 + x_2 \geq 300$

## 4: Symbolic representation of the problem
The symbolic representation is:
```json
{
    'sym_variables': [('x1', 'digital watches'), ('x2', 'analog watches')],
    'objective_function': '15*x1 + 10*x2',
    'constraints': [
        'x1 >= 150',
        'x2 >= 120',
        'x1 <= 200',
        'x2 <= 180',
        'x1 + x2 >= 300'
    ]
}
```

## 5: Convert to Gurobi code in Python
Now, let's implement this in Gurobi using Python:
```python
import gurobi

def solve_watch_problem():
    # Create a new model
    model = gurobi.Model()

    # Define the variables
    x1 = model.addVar(lb=0, name="digital_watches")
    x2 = model.addVar(lb=0, name="analog_watches")

    # Set bounds according to the constraints
    x1.lb = 150
    x2.lb = 120
    x1.ub = 200
    x2.ub = 180

    # Objective function
    model.setObjective(15 * x1 + 10 * x2, gurobi.GRB.MAXIMIZE)

    # Additional constraints
    model.addConstr(x1 + x2 >= 300, name="total_watches")

    # Solve the model
    model.optimize()

    # Check if the model is optimized
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Digital watches: {x1.varValue}")
        print(f"Analog watches: {x2.varValue}")
        print(f"Profit: {model.objVal}")
    else:
        print("No optimal solution found.")

solve_watch_problem()
```