Your DNA would not be sequenced in an assay designed to detect Covid-19 RNA.
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Both involve swabs. Could DNA be extracted from the raw swab prior to sequencing?
In theory, the raw swab would contain a mixture of DNA, RNA, lipids, and proteins. However, once the extraction process takes place, the DNA ends up discarded along with the proteins and lipids.
Just about any sample taken for analysis is going to have your DNA in it. I wouldn't worry too much about it.