First, your formula for change in working capital is incorrect. Properly,
Working capital = Current assets – Current liabilities
Δworking capital = Δcurrent assets – Δcurrent liabilities
It’s easiest to compute Δworking capital from two successive balance sheets, but you could do it with a cash flow statement if CFO were computed using the indirect method, as it will show the changes to current assets (don’t forget the change to cash: the final value) and the changes to current liabilities.
If you have inflows from financing (selling bonds or issuing stock), you still owe that money back to those who provided it (as a liability (bonds) or equity (stock)), so it doesn’t qualify as free cash flow to the firm.