Can I file tax for both F1 and H-1B partially the first year?

First, you would only be a nonresident alien for 2020 if you were an "exempt individual" in your days in F1 status, which would only be true if you had been an exempt individual as part of less than 5 previous calendar years. (For example, if you came on F1 in 2016 or later and was never previously in F or J status, you would be an exempt individual for your days on F1 in 2020; but if you came on F1 in 2015 or earlier, you would not be an exempt individual for your days on F1 in 2020.) I will assume that you have correctly determined that you were an "exempt individual" in your days in F1 status in 2020, and thus you are a nonresident alien for 2020 and thus you are exempt from FICA taxes during F1 status in 2020.

The process to get erroneously-withheld FICA taxes refunded is described on this page. You cannot get it through the income tax return filings. You should first try to get your company to refund it. If they cannot refund it, you will have to file Forms 843 and 8316 to request a refund from the IRS. Anecdotally, people have mentioned that they have still not received a refund years after filing these forms, so expect it to be a long process.