You get the W2 for all your income and withholding for the whole year. It doesn't matter if you were part time or full time. If you did not have enough withholding taken out you need to give your employer a new W4 here to increase it. Nothing you can do about the past now. You might check with the employer to see if he used the right withholding table to look up the amount each paycheck.
And the IRS is always changing the withholding tables so you have less taken out each time instead of getting a big refund at the end of the year.
You wouldn't usually receive a second W-2 when you change from full-time to part-time with the same company. If it shows that you didn't have enough withholding throughout the term of your employment, it is likely because your Form W-4 wasn't updated to change your withholding.
Unfortunately, there isn't anything that you can do about it for 2019, but you should fill out a new form for 2020 and submit it to your employer as soon as you can. You can find that form here: Form W-4 2020.
You wouldn't expect two separate W-2's from the same employer simply because your work hours changed. Have you finished your tax return? Don't assume you owe taxes until you've entered all your information and your return is completed.
If you finish your return and you find that you do owe taxes, it means that you did not have enough taxes withheld from your pay.
You can adjust your withholding at any time by filing a new W-4 with your employer. https://www.irs.gov/pub/irs-pdf/fw4.pdf