Cory,
Wow! You must be fairly close to your local TV stations if you are getting sound without an antenna on that set. Most TVs require some sort of antenna to get anything, unless they are extremely close to the transmitters. (I don't know where in Iowa you are, but I live in a fringe area 45 miles from the nearest stations and don't get much of anything on my TVs without cable.)
As to the picture problem (no light on the face of the CRT), I'd check the horizontal oscillator, output and HV rectifier first, before anything else. There could be other problems with the HV filter capacitors, flyback, etc. but since 80 percent of all troubles in tube-powered television sets are caused by weak or dead tubes, I would check the tubes first.
Beware of the high voltage on the CRT, which in color sets can run 25 kilovolts or more for a large-screen tube such as your set has. Remember to discharge the HV filter capacitors in the HV cage (and the coating on the bell of the tube, which in modern sets is the HV filter capacitor--the doorknob types went out with metal-cone CRTs like 16GP4s and the like). If you don't, you can get a nasty shock from the discharge.
Do not work on the set with the power on and the plug still in the wall socket. I know that sounds elementary, but many people have been injured or killed working on TVs that were still plugged in and
turned on.
BTW, I have never had bad luck with Radio Shack tubes. I used them quite a bit when I was working on old sets in the '70s; the tubes all worked well. These were the ones with the gold-plated pins, though. I read in another post that RS resorted to rebranding used tubes long about 1999 or 2000. I agree with that poster--these relabeled tubes are junk. Who knows how good or bad a used tube is? Caveat emptor (let the buyer beware) applies when buying old or used tubes/components as much as it does to bidding on used electronics on ebay.
BTW (2): Once you get your set working as it should, you will have a TV you can be proud of. Zeniths of 1970s vintage and earlier were very good sets and were actually made by Zenith of Chicago, not like today's troublesome "Zeniths" coming out of Korea and manufactured by Gold Star. These sets are not made by Zenith, even though the Zenith name is on them. Zenith, for all intents and purposes, went out of the TV business when GS bought them out in the '90s. The Zenith name is now only a marketing label, as is the stylized RCA logo on sets now made by Thomson Electronics.
Zenith used to be my favorite brand of TV set, but since they moved overseas, I wouldn't own one today. I purchased my present RCA XL-100 19-incher new in 1999 and have had little trouble with it to date. Heaven forbid if that set would have been a Zenith. I have a Zenith Sentry 2 19" color set in my bedroom which is eight years old and still works well, however. Perhaps Zenith TVs were still being made halfway decently in the mid-90s; who knows? Or maybe my set was an exception to the rule?
Good luck and very kind regards,