I had a much longer post about this but I think I can put it more succinctly.
I think you can prevent the Lightning from existing with a pair of highly coincidental PODs, but the bottom line in the late 1980s and early 1990s was that the league was chasing a national footprint for a more lucrative TV deal. I don't know how you can stop a bunch of businessmen from chasing that money. Even if they've built around tickets and merchandise, they're going to also want a big TV deal if they can get it.
Additionally, they wanted big-name owners like Disney and Blockbuster to increase the league's prestige. There's no way you can swap the Mighty Ducks of Anaheim and the Florida Panthers for a 1993 expansion class of the Hamilton Bulldogs and Portland Navigators.
Also, you're not going to stop the North Stars from moving to Dallas without giving them local ownership that just wasn't there at the time. The league wanted a team in Texas (again, national footprint and TV deal). It was either the Stars or the Oilers, if not both.
Let's say, though, that Anaheim and Miami and Dallas in 1993 somehow prove to be enough. The team we know as Tampa is in Seattle, with that group having gotten their financing together in 1992 while the Espositos are unable to bring in their Japanese investors. The Canadian dollar is still plummeting so the Nordiques go to Denver in 1995. Come 1996, the Jets are lost but there's no push to go to Phoenix so they move to Minnesota. In 1996, the Whalers move to Columbus to play in an abandoned aircraft hanger (seriously, that was the temporary plan) while their new arena is built.
The early in the 1990s, the NHL was planning to hit 28 teams by the end of the decade. So the late-1990s expansion is by two to hit 28 instead of four to hit 30. Bids for that round came from St. Paul and Columbus (who already have teams in this timeline), Houston, Oklahoma City, Atlanta, and Nashville. Maybe you can get Paul Allen in Portland to get his finances together, enticed by a rivalry with Seattle. I don't think the league will turn down Ted Turner's money, though, so they're going back to Atlanta. Call it Portland in 1998 and Atlanta in 1999 and we've got a 28-team league missing Tampa, Raleigh, Phoenix, and Nashville but having gained Seattle and Portland.
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With the short version out of the way, here's a longer option that's a little more creative...
If you're willing to shake things up a bit more, go back to 1972. For whatever reason, the NHL decides not to try to compete with the WHA, which means they don't go into Atlanta and Long Island for the block. It's hand-wavy; maybe they don't think they need to compete at that point, but they're coming off of thinking they needed to kill the WHL so I don't know why they wouldn't see the WHA as a threat. Anyway.
The NHL still goes to Kansas City and Landover in 1974. There's prestige to being in the US capital so they won't pass up the new arena there and they need another team to balance it out. The Scouts don't last and move to Denver in 1976.
With stability in New York and with the Miami Screaming Eagles having moved to Atlanta instead of Philadelphia (and then Vancouver, and then Calgary), the WHA is stronger in 1976. The Cleveland Crusaders are solvent, and the NHL still isn't looking to take the WHA on, so the Gunds never move the Golden Seals.
Over 10-15 years, the two leagues carve out their own geographic niches. By 1985, the WHA kind of sandwiches the NHL. It's anchored in the south, with Atlanta, Birmingham, Houston, Dallas, Phoenix, and Southern California (San Diego). Then they've got Winnipeg, Edmonton, and Calgary in the Canadian prairies. Quebec, New England (Hartford), and New York are what's left of their intrusions into traditional NHL territory, while Cincinnati and Cleveland make Ohio a WHA state.
The NHL, meanwhile, is as follows: Montreal, Toronto, Boston, Chicago, Detroit, New York, Los Angeles, California (Oakland), Minnesota, St. Louis, Pittsburgh, Philadelphia, Buffalo, Vancouver, Colorado (Denver), and Washington (Landover, MD).
Competition for players is taking its toll on both leagues. Both leagues want a big TV contract but can't get it without the other's territory. In the WHA, Southern California and New England are barely hanging on, basically propped up to keep the league relevant in Boston and Los Angeles (both of which are a stretch anyway).
Though the late 1980s the two leagues cooperate on interleague exhibitions and all-star games. A four-team "champions tournament" (made up of each league's conference champions) is scheduled in 1988 and 1989, but both years the Stanley Cup Champions find an excuse for backing out (the NHL refuses to risk their champion losing to someone from the upstart league).
Finally, after years of negotiations and rumors, the two leagues merge in 1992. As part of the merger, the New England Whalers are sold to H. Wayne Huizenga and moved to Miami while the Southern California Mariners are sold to Disney (on the heels of their hit kids movie, "The Mighty Eagles" [in the movie, the team's benefactor's name is Eagleston, as Disney wasn't going to have their team share a name with the WHA's Long Island Ducks]) and moved to Anaheim.
The 1992 NHL is as follows:
Mighty Eagles of Anaheim
Atlanta Blazers
Birmingham Bulls
Boston Bruins
Buffalo Sabres
Calgary Cowboys
California Golden Seals
Chicago Blackhawks
Cincinnati Stingers
Cleveland Crusaders
Colorado Rockies
Dallas Texans
Detroit Red Wings
Edmonton Oilers
Florida Panthers
Houston Aeros
Long Island Ducks
Los Angeles Kings
Minnesota North Stars
Montreal Canadiens
New York Rangers
Pittsburgh Penguins
Philadelphia Flyers
Phoenix Roadrunners
Quebec Nordiques
St. Louis Blues
Toronto Maple Leafs
Vancouver Canucks
Washington Capitals
Winnipeg Jets
Technically, the NHL didn't expand much, they just merged with a league that did.
Follow-up thoughts...
I imagine Birmingham, Cincinnati, Quebec, and Winnipeg would be at risk through the 1990s. Who even knows what cities would be building arenas then. Nashville and Raleigh are possible landing places but Columbus won't be as long as one of Cincinnati or Cleveland survives. Hamilton might be an option with their then-new arena. Ottawa or Tampa could build an arena. East Rutherford is probably out, with their arena outdated by then.