I need an "I got them seared into my mind/process during 3e and 4e, so I'm currently using them, but I'm desperately trying to remember how to run TotM" option.
I got into the habit in 3E/4E as well, but made a point to do TotM exclusively for 5E. It was tough at first. I have adopted my own little "house rules" that might help.
1) Describe distances in 30foot increments, either by saying adjacent, 30, 60, 90 etc or by (my method) saying engaged, close, medium, long, extreme and just ignore numbers. For ranged attacks (dis), just rename the ranges: 30/90 becomes short/medium eg
2) Players expect to be able to do things like Flank to get Adv. If PCs outnumber Creatures in an engagement, you are flanking (same for creatures)
3) Can have multiple engagements: Fighter + 3 kobolds here, Wizard+Cleric+3 Cultists there (For really crazy multiple engagement fights, I use 3X5 cards with names on them to show who is engaged where. If cards touch, they are close, if the gap is about a card in length then medium range, two cards apart then long, anything at extreme doesn't need a card. Creatures engaged provide each other with cover w.r.t. ranged attacks (ie firing into melee) period. They are circling/dancing around each other and make sudden moves. Might as well count as cover (ie penalty rather Dis)
4) When describing the scene/fights I have moved from detailed static descriptions to dynamic descriptions: "You see 3 kobolds at close range preparing to fire their slings from around the corner of the building. In front of them 2 Dargon Cultists, naked but for the blue-paint and hideous bone masks and of course the military picks they carry rushing towards you. Roll Initiative." This way, when the players do something, I can wing a little more: "You move to engage the cultists and stop them halfway between the kobolds and the rest of your party. Your swing disembowels the cultists who screams 'My Life for you my Queen!' before collapsing in steaming mess. The Kobolds launch their sling bullets (at the rest of the party) before darting back behind the building." That sort of thing. I make sure to include info about distance/cover/who is engaged, etc, but stay away from numbers (kobolds are 35' from party, but 10' from fighter and (now) lone cultist who are engaged, after the kobolds attack they have full cover from party, but 1/2 cover from fighter, etc).
5) try not to overwhelm yourself with too much detail. In example above, all kobolds operate as a single unit. At a 4E battlemat table, one kobold may not have the move to get into cover, or if a party member moves perpendicular 10' may have a shot with 3/4 cover, etc...but its not worth bothering with in TotM. Don't overtax yourself.