Welcome Eve! That's quite an interesting tiny house, and yes, it appears it was a Sunline at one time.
The serial number 278278 is correct for a 1978 Sunline. The 17 digit VIN format wasn't federally regulated until 1980, so this is the format that Sunline used prior to that. Sometimes, especially before 1976, there would be a T, C, or F in front of that number to indicate if it's a travel trailer, truck camper, or fifth wheel.
Your second picture looks correct for a 1978 Sunline as well, the orange bottom metal was on all of them. Here's a smaller scale 1978 model:
It's hard to say what model yours started out as, unfortunately that wasn't defined in these older serial numbers. If you can get a measurement tip to tip, bumper up to the tip of the front coupler, that should help. It appears to have many non-original windows added, so that complicates identifying the model. It looks like around 20', but the axles look farther back than those models would have.
As to where you'll find the serial number, there were four places on the original trailer. On a piece of paper on the wall inside the largest closet, on a sticker outside on the sidewall, under the door sill, and on the tongue. The easiest for you to find would be on the tongue, on the coupler near the jack, on the side opposite the door. It will be stamped into the coupler metal, near where it's welded to the tongue frame. It's a fairly shallow stamping and done by hand, so you may need to scrape some paint off to see it.
Depending on how the floor and door system has been modified, you may not have the inside plate. It was under the original door frame, attached to the floor. The metal tag was maybe 3" x 2" with that same serial number stamped in it.
I'd love to see more pictures of how this was customized, it looks like quite a unique trailer.
Oh, your pictures turned sideways because that's how you originally took them, likely with your phone. It seems the forum software likes to rotate them back to how you originally took it.
2007 T-286SR Cherry/Granola, #6236, original owner, current mileage: 9467.8 (as of 5/26/19)
1997 T-2653 Blue Denim, #5471
1979 12 1/2' MC, Beige & Avocado, #4639
Past Sunlines: '97 T-2653 #5089, '94 T-2251, '86 T-1550, '94 T-2363, '98 T-270SR