You're supposed to use antibiotic-free medium because they can interfere with formation of the DNA-lipid complex. I do not think this is a large effect. Transfections have worked just fine in the presence of antibiotics in my hands. As a compromise, you might add antibiotics after the transfection is complete, after 6 hours.

Make sure all your ingredients are sterile when working with them.

Also check the instrumets (e.g. pipettes): are thay clean? Has something gotten into the chanel? do they need cleaning/autoclaveing? Alternatively, use single-use sterile-packed consumables at least until you have a clean stocks frozen for further use and as backup.

If you use penicillin/streptomycin as antibiotic in your cell culture: resistance to these is quite common. To recover cells from and prevent contamination I have very good experence with the antimicrobial agent "ZellShield" https://www.minerva-biolabs.com/en/products/mycoplasma-bacteria-contamination-control/cell-culture-contamination-prevention/zellshield/ - it could be worth a try, particularly if you came across this problem more often.

Further guidance on good aseptic technique in cell culture can also be found here: https://www.corning.com/worldwide/en/products/life-sciences/resources/stories/at-the-bench/cell-contamination-containment-and-correction-good-cell-culture-practices-in-3d.html

lipofectamine 3000 works fine in the presence of antibiotics, so i suggest you to add antibiotics in medium, and instead of adding selection media it is recommended to replace with fresh medium. And to make sure your plasmid is not a source of contamination use mock control ( contain transfection reagent except plasmid).