The church recently introduced a new header on lds.org. It seems to have messed up a few things. I didn't realize it until you mentioned it, but, apparently it impacts printing, too.
You could copy the text of the talk and paste it into a word processor. You may need to paste it as unformatted text to get better results. If you're using Microsoft Word, you may get paste option buttons to help you select which paste option you want; if you're using another word processor, you might want to look for some kind of "Paste Special" option. You're still going to end up with footnotes in the text. I think the church used to have an option to turn off footnotes, but that seems to be gone now.
A very good option for your specific need (printing to hand out to class members) might be to download the article as a PDF from the Ensign (and I assume Liahona). If you click on the download button, you have a PDF option. That gave me a nicely-formatted talk as it appeared in the magazine. One talk I was studying today was 5 pages if printed from the General Conference web site, 7 pages if printed from the Ensign web article, and 3 nicely-formatted pages if printed from the PDF.
Also, many copiers in church libraries have USB connections. If you have a laptop, you could bring a cable and plug it directly into the printer (you might have to install a driver if you haven't used that printer previously) and get really nice copies to hand out. These will be much nicer than a copy of a copy that you printed at home. Many people may not have terrific printers at home and the text may be fuzzy. If you make a copy of it in the church library, it gets worse. If you don't have a laptop, maybe you can send the article to someone who does. A little advance planning could make your handouts much more legible. I don't know if any church-issued printers have wireless capability. I've seen some people ask about it in other forums. If the printer in your library has wireless, you might be able to connect from a phone or tablet to print the PDF.