To solve the given optimization problem, we first need to translate the natural language description into a symbolic representation. Let's define the variables and the objective function along with the constraints.

We have two types of plush toys: small and large. The decision variables can be defined as follows:
- $x_1$: The number of small plush toys made.
- $x_2$: The number of large plush toys made.

The objective is to maximize profit, given that each small plush toy brings a profit of $3 and each large plush toy brings a profit of $5. Thus, the objective function can be represented as:
\[ \text{Maximize: } 3x_1 + 5x_2 \]

Now, let's consider the constraints based on the resources available:
- Each small plush requires 5 units of cotton, and each large plush requires 8 units of cotton. The total amount of cotton available is 250 units.
\[ 5x_1 + 8x_2 \leq 250 \]
- Each small plush requires 10 minutes of worker time, and each large plush requires 12 minutes of worker time. The total worker time available is 500 minutes.
\[ 10x_1 + 12x_2 \leq 500 \]
- Additionally, the number of plush toys made cannot be negative, so we have non-negativity constraints:
\[ x_1 \geq 0 \]
\[ x_2 \geq 0 \]

Given this analysis, the symbolic representation of our problem in JSON format is:

```json
{
    'sym_variables': [('x1', 'number of small plush toys'), ('x2', 'number of large plush toys')],
    'objective_function': '3*x1 + 5*x2',
    'constraints': ['5*x1 + 8*x2 <= 250', '10*x1 + 12*x2 <= 500', 'x1 >= 0', 'x2 >= 0']
}
```

To find the solution using Gurobi in Python, we'll implement the following code:

```python
from gurobipy import *

# Create a new model
m = Model("plush_toys")

# Define variables
x1 = m.addVar(vtype=GRB.CONTINUOUS, name="small_plush", lb=0)
x2 = m.addVar(vtype=GRB.CONTINUOUS, name="large_plush", lb=0)

# Set the objective function
m.setObjective(3*x1 + 5*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(5*x1 + 8*x2 <= 250, "cotton_limit")
m.addConstr(10*x1 + 12*x2 <= 500, "time_limit")

# Optimize the model
m.optimize()

# Print the solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Number of small plush toys: {x1.x}")
    print(f"Number of large plush toys: {x2.x}")
    print(f"Maximum profit: {m.objVal}")
else:
    print("No optimal solution found")
```