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During a brief Instagram Live on Sunday night, the former Bachelorette, 30, filmed herself introducing Moynes, 29, to her cat, Tommy, after the couple landed in San Diego, California and headed to her place there.
"What do you think, Tommy?" she said as Moynes, a wildlife manager, expertly won over the pet.
"Alright guys, I think Tommy approves," Thurston confirmed, noting that it was Moynes' first time in San Diego.
Later on Sunday, she posted more clips and photos of Moynes and Tommy, captioning one video, "They're vibin."
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"We are still living in two different countries at the moment," said Thurston, who lives between San Diego and Seattle while Moynes is in Ontario, Canada. "We have no doubts we're getting married, but we have to figure out a few things first!"
"We're going to travel around a bit in Canada first," Moynes added. "And then we're going to check out San Diego for a potential move. It's about getting our roots in. And we're playing around with options."
Eric McCandless/ABC Katie Thurston and Blake Moynes
A few weeks later, Thurston posted an Instagram Story about the difficulties of navigating their long-distance relationship.
"I don't think people fully understand what it's like to date someone from another country and the rules behind that and the visas behind that," she said. "I think right now, Blake can only be here for 60 days before having to go back to Canada, so we really do have a lot we have to look into — and we just haven't yet — before we can plan really anything else."
Despite the logistical challenges, she previously told PEOPLE she has "never had a relationship that I've been so confident in."
"Blake and I are just very alike in the way we see the world," she said. "He's passionate, he's funny. ... Everything we've gone through has strengthened our relationship. And I'd do it all over again if it meant I'd get to be with him."