A goal by Jennison Myrie-Williams in first-half stoppage-time appeared to edge the Valiants towards a fourth win in five games in League Two, but Andy Scott's visitors launched a spirited second-half revival at Vale Park.
Paul McCallum cracked a 74th-minute equaliser as he and fellow substitute Michael Rankine roughed up the Vale defence.
Adams said: "It's disappointing, but you've got to give credit to Aldershot because they deserved their point and could have got all three.
"We looked leggy and Aldershot were probably fresher after not playing at the weekend, but it's another point and that adds to the pressure for the teams who are chasing us.
"It's well done to Aldershot and I hope they stay up. They are a terrific club and they came here and had a go.
"We will move on from this and try our best to win at Rochdale on Saturday. Our record shows we like to attack away from home and we will certainly go for it again.
"I haven't looked at the permutations, but I know the teams who are on our tail have got difficult games at the weekend. We'll keep going and remain positive."