I was just curious to know if their souls combined into one bigger soul or they exist as separate entities that can interact at will, or another possibility I haven't thought of.
They'd stay two separate souls (if snakes do actually have souls).
It’s highly unlikely that the Dark Lord’s soul piece would have merged completely with Nagini’s soul (presuming snakes even have souls - it’s never said whether animals have souls). Making a Horcrux from a living creature doesn’t mean they’re combined into one being with unified thoughts. Dumbledore says the danger of making an animal into a Horcrux is that it’s trusting your soul in something that can think and feel for itself. That means turning an animal into a Horcrux doesn’t take either of those abilities away from it - it still retains its own mind and free will.
“The snake?’ said Harry, startled. ‘You can use animals as Horcruxes?’
‘Well, it is inadvisable to do so,’ said Dumbledore, ‘because to confide a part of your soul to something that can think and move for itself is obviously a very risky business.”
- Harry Potter and the Half-Blood Prince, Chapter 23 (Horcruxes)
That means Nagini wouldn’t be “taken over” entirely by the Dark Lord, and would indicate if she has a soul, it would remain separate, not combined with the Dark Lord’s. She had a connection to the Dark Lord by his piece of soul, but she remained an individual, though one with a close tie to him.
Their connection allowing communication between them didn’t seem to usually involve their souls interacting with each other. While there’s not much information about his connection with Nagini, the Dark Lord only feels unbearable pain from close contact with Harry’s soul when trying to possess Harry, but not when Harry sees into his mind. Their minds are linked, but the Dark Lord doesn’t have any contact with his soul until that attempt to possess him. Even then, their souls are in close contact, but it’s not described as at attempt to merge their two souls together.
“Not so long ago he had one, small taste of what truly sharing Harry’s mind means to him. It was pain such as he has never experienced. He will not try to possess Harry again, I am sure of it. Not in that way.’
‘I don’t understand.’
‘Lord Voldemort’s soul, maimed as it is, cannot bear close contact with a soul like Harry’s. Like a tongue on frozen steel, like flesh in flame –’
‘Souls? We were talking of minds!’
‘In the case of Harry and Lord Voldemort, to speak of one is to speak of the other.”
- Harry Potter and the Deathly Hallows, Chapter 33 (The Prince’s Tale)
Their souls and minds clearly have a connection, but they aren’t normally in close contact, until the Dark Lord himself, not simply a soul piece, tried to possess Harry. From this, we can infer that the Dark Lord would come in close contact with Nagini’s soul (if it exists) when possessing her, but not necessarily when just communicating with her. It’s unclear exactly how they communicate over a distance, like when Nagini was disguised as Bathilda Bagshot and had to tell him she had Harry, but it seems likely that it’s not their souls merging together in any way.