What do I need (Software- or Hardware-related) to accomplish what I want to do? I am on a limited budget and don't have lots and lots of money to throw at the solution.
What you're describing is a digital signature field that you added, which does not prompt you to hand draw a signature as you've discovered.
What you can do instead is remove the digital signature field and instead use the Sign feature of Acrobat/Reader to add a hand-drawn signature. The problem is when you save the form, it get flattened, which means the form is no longer editable/fillable. An alternative is to use the pencil tool or a previously prepared stamp to add a hand-drawn signature. The problem with all such approaches is that they may not meet your needs for a signature. Using an e-signature service such as Adobe's DC eSign Services is a way to provide legally-binding e-signatures with forms, but such forms are more limited compared to a regular Acroform created with Acrobat. The point is, the best approach to take depends on your specific needs for the signature.
You will need to use an Electronic signatur pad,one of the STU series from wacom. With the aid of a thrid party program called Sign Pro Pdf to get what you are looking for.