Pure monk has more attacks with similar mobility
Rogue and monk don't multiclass very well
Rogues want a single, accurate attack. Monks want many attacks. Rogues gain a Sneak Attack die every other level. Monks gain a ki point (and thus a use of Flurry of Blows) every level. Rogues can Disengage as a bonus action for free. Monks can Disengage as a bonus action by spending a ki point. Rogues rely on Evasion at level 7 to survive area spells. Monks rely on Evasion at level 7 to survive area spells.
In the end, rogue and monk have both too much and too little in common to result in an effective multiclass. Ordinarily that would be fine - there's no need for every character to be minmaxed. However, combining rogue and monk prevents you from getting as many attacks as possible, while adding little to your mobility.
If you go Rogue 3 / Monk 5, then you won't get Extra Attack until level 8. This conflicts with your desire to have as many attacks as possible. Even if you went Monk 5 before dipping rogue, each level of rogue you take is one less attack per short rest (by spending a ki point on Flurry of Blows).
Base monk can spend 1 ki point to Disengage as a bonus action, using Step of the Wind. This costs you the two attacks you could have made with Flurry of Blows. Stunned enemies can't make opportunity attacks, so Stunning Strike also provides a (less reliable) way to escape.
Way of the Open Hand allows you to prevent an enemy from taking reactions (including opportunity attacks) when you hit them with Flurry of Blows. This doesn't reduce the number of attacks you can make, but does carry some risk - it doesn't work if you miss.
Way of the Drunken Master has Drunken Technique, which removes the risk entirely. You gain the benefit of Disengage when you use Flurry of Blows - no hit required. It also increases your speed by 10 feet.
If all that mobility isn't enough, you could take Mobile:
When you make a melee attack against a creature, you don't provoke opportunity attacks from that creature for the rest of the turn, whether you hit or not.
Unlike all of the other monk options, the Mobile feat works without spending any ki points. If you find yourself engaged by multiple enemies, you can make one attack against each one in order to escape safely - you don't even need to hit. Mobile also increases your speed by 10 feet.
Unfortunately, you can only use one maneuver per short rest with the Martial Adept feat alone. If you prioritize mobility, then the Mobile feat is superior to any maneuver (including the mediocre Evasive Footwork). If you prioritize attacks hitting, then +2 Dex is your best bet.
The Riposte maneuver allows you to make a melee attack as a reaction when an enemy misses you with a melee attack. This grants you one additional attack per short rest, but if your goal is to increase your raw number of attacks, I would instead recommend the Sentinel feat.
Swashbuckler either comes online late or delays your Extra Attack. For optimal attacks and mobility, I recommend a pure Way of the Drunken Master monk without Martial Adept. If Drunken Technique doesn't give you enough mobility, the Mobile feat should fill any gaps. If you desire more attacks, the Sentinel feat will almost certainly provide you with more per short rest than Martial Adept.
If you want a Swashbuckler with additional attacks, I'd recommend a dip in Battle Master fighter instead of monk. Three levels gets you maneuvers (with way more maneuver dice), as well as a Fighting Style (Dueling or Two Weapon Fighting), medium armor, shields, and Action Surge1. Going to level 5 gets you Extra Attack.
1 Gives you an extra Sneak Attack if you use Ready. You can only get one SA per turn, but multiple in the same round.