Honestly the only way to determine if he has ADHD/ADD and sensory processing disorder is to have him evaluated with a Neuropsychological examination/evaluation by one of the following: pediatric neurologist; child psychiatrist; developmental-behavioral pediatrician; child psychologist. Any of those professionals can diagnose him. At nearly 3 years old, it's not impossible to diagnose him. It sounds like he might have some other issues going on in addition to or instead of ADHD so an evaluation wouldn't be uncalled for.
EDIT [I meant to add that if a child has SPD, it's very common for them to have it co-diagnosed with another neurological disorder. It's very rare actually for there to be SPD completely on its own, just like it's very rare to have Nonverbal Learning Disorder completely on its own without Autism or ADHD or something else.]
One reason I would encourage you to get one done sooner rather than later is that especially if he has Sensory Processing Disorder, and if there are other issues in addition to ADHD, it's far better to get services in place and started now rather than later. Early intervention is key. He might need Occupational Therapy which is so much more effective the earlier it's begun.
Early diagnosis does NOT mean he would automatically be put on medication, and please don't assume that medication would be so that you can "deal with him." Medication should always be so that life is easier to cope with for your son, and when that happens life is easier for everyone around him. Don't let people that are anti-medication scare you, okay? Besides, the right doctor will not push you to put him on medication and they'll suggest therapies you can do at home, dietary suggestions, and holistic remedies before trying medications AND in tandem with them.
I know this because I have a daughter with ADHD and SPD. I also have a daughter that has Autism, ADD, and SPD and a host of related issues.