Can I sue the manufacture of a new dishwasher because I lacerated my finger?

If installing dishwashers is your job, you have a worker's compensation claim to be filed through your employer's HR department.
If installing dishwashers is NOT your job, prior to filing a lawsuit against the manufacturer, it may be possible to settle the case with the manufacturer's insurance carrier.  Notify the manufacturer in writing of your personal injury claim and obtain its insurance carrier information.  Notify the insurance carrier in writing of your personal injury claim.
When you complete your medical treatment and are released by the doctor or are declared by the doctor to be permanent and stationary, which means having reached a point in your medical treatment where no further improvement is anticipated, obtain your medical bills, medical reports and documentation of wage loss.  Your personal injury claim filed with the manufacturer's insurance carrier should include those items.
Compensation for the medical bills is straight reimbursement.  The medical reports document your injury and are used to determine compensation for pain and suffering which is an amount in addition to the medical bills.  Compensation for wage loss is straight  reimbursement.
If the case is settled with the manufacturer's insurance carrier, NO lawsuit is filed.
If you are dissatisfied with settlement offers from the insurance carrier, reject the settlement offers and file a lawsuit for negligence and strict liability against the manufacturer.  Negligence is the failure to exercise due care to produce a product that is not defective.  Strict liability imposes liability whether or not due care was exercised.  Negligence and strict liability are separate causes of action (claims) in your lawsuit.
If the case is NOT settled, your lawsuit against the manufacturer must be filed prior to the expiration of the applicable statute of limitations or you will lose your rights forever in the matter.
If the dishwasher came directly from the manufacturer, your personal injury claim is only against the manufacturer.  If you bought the dishwasher at a store, the store is also liable  for negligence and strict liability even if it could not have known the dishwasher was defective.
The same discussion with regard to the store and its insurance carrier would apply regarding filing your personal injury claim and if the case is not settled, filing a lawsuit against the store for negligence and strict liability.  The same discussion regarding the statute of limitations is also applicable.
You would only file one lawsuit naming both the manufacturer and store as defendants.  If the case settles with one but not both parties, only name the party (manufacturer or store) with whom the case has NOT settled as a defendant in your lawsuit.