Great question @matthew K. , here's another option you have. Instead of HELOC, why not simply refinance, and use the equity to purchase another property. You don't necessarily have to sell, if you don't want.
Also, keep in mind, there are great benefits as an investor of income properties, you can write off, depreciation costs, and if, it doesn't rent, you write that off as well. Disclosure, I'm not a tax advisor, but speak to a local CPA.
What my investors do, is they locate nice neighborhoods, and buy the ugly duckling of the house, spruce it up and rent it out or sell it.
Anyhow, hope this helps.
@Matthew K. there are tax benefits to either strategy.
If you've lived in your old house for 2 out of the last 5 years, you can exclude up to $250k ($500k if you're married) of the gain from your taxable income.
If you use funds from a HELOC, you tap into the equity of your old home, pull out money tax free, and invest it. The HELOC interest would be deductible.