Can I claim full monthly cell phone expense if I only work part time as a self employed contractor (1099-NEC)?

No, you can only claim the portion you used for business. You will have to allocate expenses between personal use and business use. For example if you had $100 for cell phone expenses and you used 20% of the time for the business and 80% for personal usage, then you can take a deduction for $20 for business use and the remaining $80 cannot be deducted since its for personal use. 

If you have an expense that is both personal and business, you can only deduct the amount of the expense that can be directly allocated to business use.  If audited, you must have some type of reliable method to determine the cost that you allocate to the business, otherwise the IRS can deny the expense.