Hiding whois information on GoDaddy registered domains

There is no way around this. If you want to use godaddy as your registrar that's the price you have to pay for private registration. If you can get a domain name with private registration cheaper elsewhere, and cost is an issue to you then register your domain at the cheaper registrar.

From this question asked previously about entering fake information when registering your domain:

ICANN (not the domain registrar) requires that all information in your registration be valid.

If any dispute arises (see the ICANN Uniform Domain Name Dispute Resolution Policy rules here) you will be contacted via the means specified in your domain registration. Notice that section 14 of the rules is a section that defines what happens as part of a 'default' (in other words, they can't contact you): They'll proceed with a judgement, and you won't get a say in the proceedings.

ICANN has the power to take a domain from you and give it to somebody else.

So yes, it's important that you include valid information in your registration information.

For a comparison of a what a private domain registration looks like (compared to a regular domain registration) see this: http://www.domainsbyproxy.com/popup/whoisexample.aspx

Since it hasn't been mentioned so far: At least EU citizens are protected by GDPR. From an article on TNW:

If you order from the UK or any other EU country, you are not required to pay to make your data private in the WHOIS database.