There are no special difference to working in the public sector.
You may be having too little withheld. If you just used the new form W-4 for the first time, your withholding might be different (maybe your old job was relying on a W-4 you filed before the 2018 tax reform and never updated.)
If you have more than one job or you are married and your spouse works, you need to complete new W-4s for both jobs at the same time--whenever you update one, you must update them all. Follow all the instructions on all 4 pages of the W-4, not just the front page.
When you use the estimators, are you taking account of pre-tax deductions for medical expenses and your retirement plan? It depends on how the estimators are programmed; if it asks for your gross taxable income and then asks for pre-tax deductions, make sure it is asking for all your pre-tax deductions.
You can also use the IRS tax estimator, it is very accurate, but you need at least one or two pay stubs from your new job so it can calculate the future withholding.
https://www.irs.gov/individuals/tax-withholding-estimator
