Not meaning to parrot prior posts, but as a matter of tossing my thoughts in for reiterative purposes if nothing else, I offer the following comments:
My first thought, when I read about how your temperature ***** light is behaving, is exactly as HemiKiller stated. I suspect the incorrect style sending unit has been installed previously. Removing it and testing it per HemiKiller's suggestions is the exact approach I would be taking and suggesting (Well done, HemiKiller, again).
Using a thermal temperature gun reading device and seeing the difference in coolant temperature between the upper and lower hoses was an excellent move. It helps tell the larger part of the story. I avoided doing needless repairs due to an oversensitive temperature gauge on our 1969 Shelby GT500 by using an infrared temperature gun to verify the temperature being as high as reported by the temperature gauge (it was at 180 degrees, whereas the oem gauge showed the temp at the upper end of the "normal level. The new sending unit let the gauge show a temp reading at the slightly lower 12/3 of the normal range on the same oem gauge. I have been a raving fan of verifying a problem prior to beginning repairs from my earliest days as a repair technician, but the infrared temp gun bolstered that approach.
There have been some other excellent thoughts and suggestions also proved by other members of the forum for whom I have tremendous respect. In order to help you acquire the needed manuals and schematics suggested, I offer the following link to the Forel Publications site where you can find some excellent manuals in PDF format that of excellent quality.
This link is specifically for 1973 Mustangs and Cougars. Going up one level shows additional years of manual coverage.
https://www.forelpublishing.com/clickbank/index.html#1973
It is your good fortune you have a 1973 Mustang Convertible for a few reasons. First of all, Lynda and I also have a 1973 Mustang Convertible with a 302, C4 Auto Tranny, and 2.79:1 rear axle ratio open differential gear set. It was literally a "Barn Find," and when we purchased it a few years ago it had less than 20,000 original miles on it. Despite our adding Classic Air A/C, LED taillight panels (
VintageLEDs.com), and installing a Dual Electric Cooling Fan system, the Mustang is largely unmolested, where the upgrades performed were done with an eye on excellent workmanship. So, in addition to having excellent documentation from Forel, we have what is likely a close twin to your vert for comparative system purposes.
But wait! There's more! Although in 1972 the Forel electrical schematic manual began to provide far improved and complete information as compared to the preceding years of electrical schematics, I have found the 1973 Forel Color Electrical Schematic manual to be as close to perfect as possible. More important, in 1973 the Ford Shop Manual, Volume VI, provides truly excellent and complete emission control system and engine emission control related vacuum hose "Calibrations" (aka schematics), whereas in the prior years the Shop Manuals were devoid of that level of information. There are, however, two vacuum calibrations missing from the 1973 vacuum routing schematics, but that affects only the 351C 4v engine configurations, so you are not adversely impacted by the absence of those calibrations. None-the-less, I will include a file attachments for the missing calibrations so you have it on hand in case you ever need that info.
I have some snippets of the Shop Manual and/or Electrical Schematics Manual I want to share with you. Some of the snippets are annotated by me to bring out some clarification I felt might be helpful to myself or others. Despite providing these snippets I encourage you to license and download a full set of manuals, to include the Shop Manual and Color Electrical Schematics, so you have properly licensed and complete documentation on hand for your Mustang. The pricing is very reasonable, actually on the very inexpensive side, especially considering how much information is packed in those PDF files.
If you run into a pinch or need to bounce things off the wall, I am sure my fellow enthusiasts would join me in welcoming your questions or concerns. That said, I know that you can count on me providing information, photos, videos, etc., as needed to help you out. And speaking of videos, for what it is worth Lynda and I have provided a lot of files covering various subjects on the diagnosis, maintenance, and repair of these First Generation Mustangs. They are found under my YouTube Channel ("Gilbert Hale"). If you need clarification that is best served by using a video to show the answer or direction, it will likely end up being added to our channel. If not, just post the question or concern.
https://www.youtube.com/@gilberthale7777/videos
Following are some links for the documentation snippets I referred to above. They are usually too large to attach to posts or email, so I put them on my Google Drive to share as needed. I hope you find them useful:
For the entire folder:
https://drive.google.com/drive/folders/1TdZ-HG5Y2dpOFR1iNdDFOkvE9DiANibl?usp=drive_link
For individual files:
https://drive.google.com/file/d/1B7km6-h8YTZPpc1_aMLqaov_qAK5dPnV/view?usp=drive_link
https://drive.google.com/file/d/1KbgNEsC3bZ7t1GWSjURODFda1IgTDhtq/view?usp=drive_link
https://drive.google.com/file/d/1x2SdykXlUfQTPMQdzhG8X1SxXYVpMfQO/view?usp=drive_link
https://drive.google.com/file/d/1FdPOb1qkf9E-9GlBpPtFxOJITMK0DN1n/view?usp=drive_link
https://drive.google.com/file/d/13z5Lrpv3PaWy-uoaKoV2xZzuw8EFms4d/view?usp=drive_link
https://drive.google.com/file/d/1gO2kbjEmqHJj4ITwACIqo1TUxZOWXNAi/view?usp=drive_link
https://drive.google.com/file/d/18P9oiebm2nUcYjQ-RFJA9XWTcB8VhOod/view?usp=drive_link
https://drive.google.com/file/d/1-foyzcIRVLOT3ZmdroMukADeWhG-B-CY/view?usp=drive_link
https://drive.google.com/file/d/1qzTAmUkLPVvzHzCpeh61NskeW-oLN11C/view?usp=drive_link
https://drive.google.com/file/d/1fA7YK2vV5D-FAh_uHKynktF-lx1xOIzC/view?usp=drive_link
https://drive.google.com/file/d/1D2hBFNmQOMtGQByZg6KJbzwCc93bTo3M/view?usp=drive_link
Finally, with respect to your instrument panel bring laden with ***** lights instead of analog gauges, both of our 1973 Mustangs (Convertible and Mach 1), were also equipped with the ***** light based instrument panels. Rather than try to track down oem analog gauges for the Instrument Panel and the 3 gauges to be mounted in the center vertical bezel, I opted to purchase and install the Dakota Digital VHX Analog Gauge panel. We got ours with red back-lighting and Carbon Fiber trim. Following it their web site for the 1973 VHX kit. You can navigate around to look at their digital display panel if you prefer that look. I have also attached some photos pf you the Dakota system looks installed on out Mustangs.
https://www.dakotadigital.com/index.cfm/page/ptype=product/product_id=922/category_id=434/mode=prod/prd922.htm