An organic injection solvent like Methanol, ACN, IPA... will sharpen the peak
No.
If your molecule loves Methanol, fine! Once injected onto the column it is just going to sit there until the gradient composition (%organic) allows it to be solubilized and thus elute.
No.
Remember, chromatography is a series of equillibration steps of the analyte between stationary and mobile phase. If your injection solvent is significantly different from the (initial) mobile phase, there is a severe disruption of the equillibrium, because initially the analyte "sees" only the injection solvent and not the mobile phase. This can lead to all sorts of peak broadening and peak shape issues.
Using pure methanol or the like as sample solvent in reversed-phase is a no-go from a chromatographic point of view - but sometimes you don't have much options because of solubility issues.
Generally, you cannot predict if you'll see any artifacts from using a certain solvent - it heavily depends on all sorts of variables like column dimensions, injection volume, dead-volume of the system, flow-rate etc.
The best thing to do from chromatographic theory is to use the mobile phase (in case of gradients: the initial mobile phase) as sample solvent, so there will be no equillibrium issues.
The best thing to do from a practical point of view is to use a weaker (in terms of elution power) sample solvent in order to "compress" the peak at the head of the column. In the case of reversed-phase this means using a solvent which is more aqueous than the initial mobile phase. In the case of HILIC it's the other way round, you should use more organic in the sample solvent, because here water is the strong eluent.
If you MUST use a stronger solvent, limit the injection volume as low as possible a warch out for any peak shape issues.
An remember, organic/aqueous ratio is not the only possible difference, pH and salt concentration differences might also have influences (especially in HILIC).