Hampshire were struggling at 62 for 5, with Leach taking 3-9 as they trailed Somerset who were bowled out for 136. Kohler-Cadmore top-scored with 63, but Dawson and Abbott ran through the Somerset batting line-up taking 4-28 and 4-37 respectively, leaving Hampshire trailing by 74 runs.
Kyle Abbott and Liam Dawson claimed four wickets each as Hampshire boosted their bid for runners-up spot in the Vitality County Championship by skittling nearest rivals Somerset for 136 on the opening day of the final Division One match at Taunton.
Seamer Abbott returned 4 for 37 and left-arm spinner Dawson 4 for 28. Only Tom Kohler-Cadmore, with 63, offered much resistance after the hosts had won the toss and elected to bat first on a dry surface, expected to take spin.
By the close of a rain-interrupted day, Hampshire had struggled to 62 for 5 in reply, Toby Albert unbeaten on 38 and Jack Leach taking 3 for 9. But, having failed to claim any batting points, Somerset now have to win the game to overtake their opponents and finish second behind champions Surrey.
The home side's batting problems began with the fourth ball of the second over when Tom Lammonby, promoted from No. 3 to open with Archie Vaughan, fell lbw to Abbott without scoring. Soon it was 4 for 2 as Andy Umeed was caught by wicketkeeper Ben Brown off the same bowler for a second-ball duck.
Tom Abell, on 5, survived a confident appeal for a catch behind off Mohammad Abbas, who then bowled him through the gate for 10 with the total on 29. Kohler-Cadmore's response was to launch his innings with four boundaries in the space of five balls.
Vaughan had recovered from an uncertain start and was unbeaten on 19 when rain interrupted play with Somerset 52 for 3 from 14 overs. An early lunch was taken at 12.15pm and the action resumed at shortly after 1.20pm, with ten overs lost.
Hampshire immediately introduced Dawson to the attack and he quickly found some turn to pin Vaughan leg before for 29 with his sixth delivery to make it 63 for 4. With only a single added, James Rew edged Abbas to second slip where Albert snapped up a low two-handed catch. Kasey Aldridge went 22 balls without scoring before trying to sweep successive deliveries from Dawson, the second of which saw him fall lbw.
Lewis Gregory helped an increasingly restrained Kohler-Cadmore add 27 for the seventh wicket before being struck on the back pad by a delivery from Dawson and departing to another leg before decision for 16.
At 98 for 7, Somerset were in disarray. But Kohler-Cadmore stood firm to record his first Championship half-century of a red-ball season heavily curtailed by participation in the IPL. He had faced 85 balls and struck five fours and a six over midwicket off James Fuller.
Leach looked to be lending decent support when caught at slip for 9 aiming a big hit off Dawson. When tea was taken at 3.30pm, the scoreboard read 128 for 8.
Kohler-Cadmore did his best to farm the strike at the start of the final session and the scoreboard ticked slowly to 136 before he was caught behind driving at a ball from Abbott. Last man Shoaib Bashir, who had survived being bowled first ball because of the distraction of Abbott's towel flying to the ground as he released the delivery, quickly became the fifth lbw victim of a sorry batting display lasting just 53.5 overs.
Alfie Ogborne had battled away for 24 balls to score 1 not out and the young left-arm seamer lifted Somerset spirits by having Fletcha Middleton caught at second slip off the eighth ball of Hampshire's reply. It would have been 0 for 2 had Aldridge not spilled a routine slip catch offered by Albert off Gregory in the third over.
With the total on 13, Nick Gubbins edged Gregory to first slip where Abell took a good catch low to his right. Somerset were back in the game when James Vince, having hit Vaughan for six over deep square, aimed another big hit at Leach and skied a catch off a leading edge to Abell at backward point.
Leach struck again when Tom Prest drove a catch to extra cover and then bowled Dawson between bat and pad. For the first time in a Championship match, Somerset had picked both Leach and Bashir, bowling the two England spinners in tandem towards the end of the day.