I did a house in Dennis, kitchen, 3 baths, and a media center all inset, about 4 years ago. I was concerned, house is two blocks off the bay, but clients existing house across the street had an inset kitchen for over ten years without issue. The new home had central air but the older one did not. Both were vacation homes that saw a lot of use.

The new kitchen was done with a high end brand and I checked with the manufacturer who said they supplied plenty of insets in the are. (The baths were all done in a semi-custom with a slightly wider reveal so less worry)

Due to construction problems the kitchen cabinets sat un-acclimatized from March to August. I was really worried. There were adjustment issues to start bit most were due to installation issues. We replaced one appliance panel that had a very wide top rail which swelled noticeably.

All that said any maker will void warranty if climate is not controlled. The mfg replaced the panel at my request knowing it was NOT a warranty issue and sent a tech to help with the adjustments but at the price point I would expect a little accommodation. If there were multiple replacements needed that would have become an issue.

NOTE, some makers with tight reveals offer an increased reveal for situation like this at no charge. I didn't know about it at the time but was later informed they don't often use that on the Cape.

Two notable things you mentioned were that it's "small", and you "don't have a large budget". Inset typically is priced higher than frameless cabinets, and when it's not priced higher, the quality may not be there and you have more of a chance of getting the sticking you are worried about.

But more than that, for a kitchen that is small, keep in mind that you will lose valuable inches with inset that you would otherwise have with frameless. In particular the uppers and the drawers. A 12" upper frameless cabinet will only have 10-1/2" or so of interior usable depth. Look at your dinner dishes and see if they will fit. Look at your glassware and see if you will be able to have more than two glasses deep in the inset cabinets. An inset utensil drawer may not be wide enough in the interior to fit much utensils. Make sure to go to a kitchen showroom and look at the inset drawers to see what I mean (there are also pics on the internet if you google "inset drawers vs. frameless" or something like that). If you have a large kitchen, inset would be very nice. But with a small kitchen, and the usable space factor, I would choose frameless cabinets over inset without hesitation.

My preference for maximum storage and function in small kitchens is frameless first, inset second, framed traditional overlay third (rarely acceptable by clients), full overlay framed last. I've done very small kitchens where inset cabinets provide more storage than any other option including frameless. There is no hard and fast rule.

The available storage for inset cabinets will be equal to or greater than a comparable framed overlay cabinet. In small kitchens they can be almost as good as inset "when done well"

Wall cabinets for inset are generally between 13 and 14" standard- no loss there in depth even in semi custom brands. There is a loss of 1-1/2" in height- that can matter a little sometimes.

The places where any frameless cabinets shine is in drawers. A framed cabinet (inset OR overlay) will have 1-1/2 less usable width than a frameless.

Height of drawers can be a bigger issue and is what to watch for. Many makers have a taller top drawer for insets than a typical overlay cabinet.

"when done well" That means:
1) that as many cabinets are "combined" as possible with stiles specified as 1-1/2". Combine two cabinets and gain 1-1/2" width for drawers-every additional cabinet gains another 1-1/2" Besides combined cabinets look better AND are available in some semi custom brands.
2) blind or dead corners use smaller clearances since they can- since the door is inset 7/8" is gained in each corner (often blind or dead corners are the most efficient solution in a small kitchen)
3) when possible eliminate intermediate rails on multiple (3 and 4) drawer cabinets to gain height and configure drawer heights to optimum. This one is not always available at the lowest price semi custom-have to go up a level but can be a no cost option. There is a way around it in a brand that doesn't offer it with field install but that is another topic.

Price: some brands charge 10% more for inset than framed overlay since all else is the same a typical kitchen is increased by 5-8%, some brands charge the same for overlay and inset, some charge 10% less for frameless but again only the boxes so 5-8% swing- then comparing a frameless brand to a framed brand, well that gets complicated but again there is no simple answer.

Jakuvall, I respect your knowledge, and please believe I am not trying to debate you, but for the OP, I want to make sure she has a realistic understanding. When you say, "Wall cabinets for inset are generally between 13 and 14" standard- no loss there in depth even in semi custom brands". I am sorry, I have not seen inset uppers in 13" or 14" depth unless they are custom and more costly. In fact, there have been posts on this forum from time to time about people realizing their standard 12" deep inset cabinets don't fit their dinner plates. The OP should go to a kitchen showroom for herself and see what is available in inset cabinets in her budget, and look at depths of uppers, and widths of drawers. It is useful to bring with you a dinner plate and a couple of your glasses - so helpful in making up your mind. Don't be bashful about bringing your stuff - people do that all the time.

Many people with inset cabinets do opt for both their upper cabinets and base cabinets to be deeper than the standard 12" and 24" respectively. If you go deeper on the base cabinets, your countertop will also need to be deeper than 24", which is actually an excellent choice in kitchen design - more usable counter space. But that may eat up the aisle space in your small kitchen - hard to say without knowing your kitchen dimensions. In any case, those changes will cost more, and the OP has said she has a small budget.

Jakuvall also mentions something I had forgotten about inset - those stiles. Having lived with inset cabinets many years ago, I had cabinets where those stiles were in my way every time I opened the cabinets. With frameless no need to deal with a stile smack dab in in the middle of my cabinet. I understand that you can get inset cabinets now without stiles, but I believe it's in the more costly lines.

I just did a 30-second search, and came up with these threads. There is more info out there. Cabinets are costly, and it is worthwhile to spend some time on research.

Thread from new owner of inset and the upper cabinet depth

Thread asking about 12" deep upper inset cabinets

Thread about fitting range into inset base cabinets

I carry four brands that offer inset: Eastman Street 13" walls prices around Shenandoah, Showplace- 13 and/or 14 your choice prices around Medallion Platinum (which also has 13), Mountaineer 13 price below Omega or Crystal, and QCCI 13-1/2 expensive. All but Eastman will combine cabinets, none have center stiles on 36 or narrower cabinets, only Mountaineer and QCCI will omit intermediat rails. For the others I order a base cabinet with no doors and loose drawers which we field install.

The two smallest kitchens I've ever done were both done with insets because it worked better, a lot better.

I always encourage folks to get the look they want AND maximum efficiency. I've been measuring people's "stuff" long before it was mentioned on GW. Take pics of it all and map it out for them. I very often suggest frameless but not always. Also often price puts folks into framed overlay. There are less expensive decent framed cabinets than comparable frameless unless you look into RTA's

The bottom line is does the kitchen work for the client. There is no hard and fast rule.