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If invoices are being automatically marked as paid, it's possible your Automatically apply credits option is turned on. This feature will apply payments to new invoices and mark them as paid if the customer has a credit balance on file.
Here's how to manage this setting:
Use your Gear icon, then go to Account and settings.
Access the Advanced screen and click to its Automation section.
Turn your Automatically apply credits option on or off.
Select Save, then Done.
In the event an invoice was incorrectly marked as paid by your Automatically apply credits setting, you can delete its applied payment to mark the invoice as unpaid:
In your left navigation bar, go to Sales, then locate the invoice and open it.
Click your 1 payment link at the upper right, above its PAID payment status.
Hit your Date link.
From the Receive payment screen, choose an appropriate invoice.
Select More, Delete, then Yes.
I've also included a couple detailed resources about working with invoices and payments which may come in handy moving forward: