IRA conversion is not subject to FICA and neither is distribution; both are subject to Federal income tax for sure, and maybe state (it depends on your state).
(I assume you are a US 'resident' for tax purposes, otherwise this is a good deal more complicated; see Taxation of traditional to Roth IRA conversion for non-resident !)
But you need to remember withholding is not always the same as tax.
For FICA, the actual tax is a flat rate on amounts paid by the employer (with some exceptions the employer knows about), so the withholding is almost always correct and you don't have to do anything further. There is one exception: Social Security tax only applies up to a yearly limit called the 'wage base', which is $137,700 this year but changes with inflation; if you are paid by two or more employers and the total exceeds the wage base, more SS tax is withheld than you actually were required to pay, so you can claim a refund of the extra on your income tax return; see the instructions for 1040 line 11 (note the webpage is still for 2019, when the wage base was lower). Note Medicare tax does not have this limit, and not only continues but increases at higher incomes, and that part is affected by multiple sources or joint filing, so withholding may not be correct and you may have to pay more.
For income tax on wages -- Federal for sure and most states -- withholding is an estimate. You must file a return shortly after the end of the year -- normally by April 15 (or next business day), but you can get an extension to Oct. 15 (ditto), and this year April 15 was moved to July 15 because COVID -- on which you compute your actual tax (many people have software or a preparer do this), and if your withholding was not enough you must pay the difference, in some cases plus a penalty, but if your withholding was too much you get a refund.
For income tax on an IRA conversion (OR a distribution, other than a qualified Roth distribution which is tax-free) withholding is not really even an estimate. If you do the conversion as a direct trustee-to-trustee transfer there is no withholding. You still owe the tax, and depending on the rest of your tax situation (other income, deductions, credits, and status) you may need to make estimated tax payments during the year to avoid a penalty, or you may be able to wait until next April 15 to pay.
OTOH if you do the conversion 'manually' -- i.e. you first take the money from the trad IRA and then within 60 days deposit it, identified as a conversion/rollover, to the Roth IRA -- there is flat 20% withholding. That doesn't mean your tax will be 20%; it might be more, or less. Only the withholding is 20%. But it means to actually convert the complete amount you must 'front' the 20% cash to replace the amount withheld. Then during filing season you might get some of it refunded. Conversely depending on the rest of your situation you might owe more than the 20% and might need to make estimated payments.
Alternatively, if you use the 20% withholding on non-direct, or any other money taken from the source trad IRA, to pay the tax, that money is not considered converted but rather distributed. That means it is subject to normal income tax, plus if you are under 59.5 there is an additional 10% tax unless certain special cases apply (and from your description they probably don't). So try to avoid doing that.