There's likely an issue with the EDID handshake between your new Roku Premiere and the HDMI adapter. Some have DIP switches that can alter the information provided, so you might be able to make them compatible. If there are no switches, then they probably can't be make to play together.
Depending on which model of Roku 3 you have (there were two, one released in Mar 2013 and one Apr 2015), it's possible you're going to need to upgrade it in the not too distant future, as many online providers stop supporting devices once they've reached "old age". Roku still officially supports both versions with OS updates, but that could stop as well within a year or two. Users have reported success with newer Roku players and HDMI converters, so it's possible you might only need to replace the converter.
Just as a reminder, that converter is only giving you SD output (~480-525 lines) compared to HD (720, 1080 or 2160 lines). You're missing out on some gorgeous content when you're only watching in SD. While new TVs certainly aren't what everyone might consider inexpensive, it might surprise you what you can get for a few hundred dollars. I think the last TV I picked up (a Vizio) was about $240, it was 40" (I'm certain that's larger than what you have now) and it was a 4K/Dolby Vision capable set. My Roku Express 4K+ looks great on it.
Dan
Roku Community Streaming Expert
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